A model for structural safety using the extreme value approach

A model for structural safety using the extreme value approach Shunsuke

Baba

Department

of Geotechnical

Engineering,

Nagoya University, Nagoya, Japan

Kenji Nakagawa Department

of Civil Engineering,

Gifu University, Kagamihara, Gifu, Japan

Masao Naruoka Department

of Civil Engineering,

Nagoya University, Nagoya, Japan

(Received January I9 79; revised October I9 79)

A new method of evaluating the safety of a structure using the maximum mean largest value concept is introduced. The paper is characterized by the following points: The maximum probability of failure is defined by employing the upper bound of load and the lower bound of resistance, that is, the structural safety is evaluated under the worst-state both for load and resistance. This is a distribution-free approach to the concept of structural safety, and the probability of failure is calculated without assuming the probability density distributions of load and resistance. Three new characteristics are introduced so that the character of the data is strongly reflected on the estimation of the maximum probability of failure. A numerical example is given using the test data.

Introduction A structural safety evaluation by a new method of the extreme value approach is proposed by applying the maximum mean largest value concept originated by Gumbel for load and resistance. This is a distribution-free approach to the concept of structural safety. In the structural reliability analysis introduced by Freudenthal,rz the probability of failure is calculated by assuming the probability density distributions for load and resistance. The assumption of the type of distributions is a weak part of reliability theory, because the probability of failure is strongly influenced by the distribution assumed, as was pointed out by Ang et al. 1 It is almost impossible to say positively, for instance, that ‘This distribution must be a normal distribution’, based on data which are limited in number. The distribution may or may not be a normal distribution. In any case, the intuitive assumption is not only devoid of rationality, but is also directly related to the underestimation of the probability of failure. Ang et aZ.2 proposed a so-called extended reliability theory and tried to evaluate the variation of the probability of failure caused by the intuitive assumption of the distribution. They introduced objective and subjective uncertainties to decrease the variation of the probability of failure, but the concept lacked mathematical rigour and they have not extended the theory. 0307-904X/80/040275-11/$02.00 0 1980 IPC Business Press

The method proposed here follows Gumbel’s concept of the maximum mean largest value.7,8,1r The concept itself has seldom been used in engineering problems, while his asymptote concepts have been widely employed. The unpopularity of the first concept is attributed to the following defect: In the normal distribution, the deviation corresponding to a probability of l/1000 is equal to three times the standard deviation from its mean, while in case of the maximum mean largest value, the deviation corresponding to the same probability is equal to 32 times the standard deviation. Although Gumbel’s concept of the maximum mean largest value was deficient in some aspects, the authors are interested in the concept because: the maximum mean largest value is a distribution-free technique and can be calculated without substituting the established statistical distributions such as a normal distribution for the probability density distribution of the data. The distribution maximizing the mean largest value is found amongst all distributions which have the same characteristics as the data. The authors have considered that the evaluation of structural safety for the worst state will be realized by applying this concept for both load and resistance. The authors first tried to improve the overestimation of the maximum mean largest value by increasing the number of constraints from

Appl. Math. Modelling, 1980, Vol 4, August

275

A model for structural

safety using the extreme value approach:

two to seven,4 and extended the concept to the problem of the probability of failure by defining the modified maximum mean largest value.5T6 The extreme value approach always gives an extreme state of structural safety, because the concept is based on the minimization process of the variational methods. The concept is also practical, because each datum is employed effectively and the design for the worst state is performed.

S. Baba et al.

population are made equal to those of the data. The probability density of the population is given by f(X), and its cumulative function by F(X). Although it is usual to treat F(X) as a function of X, X is here regarded as a function of F, as the distribution F is unknown. That is, a notation X(F) is used, following Gumbel. The mean largest value of a sample of size N, FN, is expressed as follows by using the unknown function X(F):8 1

Maximum

yN =

mean largest value

The maximum mean largest value introduced calculated as follows:

by Gumbel is

(1) More than two characteristics including the mean and the variance are calculated from the data of size rz. (2) All possible populations, which have the same characteristics as the data and have individual distributions, are imagined. (3) Sets of a sample of size N(>n) are repeatedly extracted from each population, and the largest value of each sample is imagined. (4) An average value of the largest value is defined as the mean largest value. The mean largest values are calculated for all possible populations which are imagined in step (2). (5) The maximum of the mean largest values is calculated based on the variational principle, and is called the maximum mean largest value. Let each value of the data be a random variable called x. The mean and the variance calculated from the data are expressed as 2 and sz, respectively. The normalized value of x is expressed as X, that is, X = (x - 2)/s,. The maximum likelihood estimation procedure is introduced and the mean and variance of the

s

NXFN-‘dF

0

Probability of failure is defined as the ‘probability of a situation where load is greater than resistance’. One of the most practicable ways of calculating the maximum probability of failure is to introduce a special distribution function for load and resistance, which has a pulse part represented by the b-function. That is, distribution for load has a pulse part at the maximum, and distribution for resistance has a pulse part at the minimum. When the pulse part of the load-distribution is located just on the pulse part of the resistance-distribution, probability of failure is evaluated approximately as a half of the area of both pulse parts. The pulse part is regarded as an unknown parameter, by which the extreme situation is created based on a variational principle. The definition of Y, has no connection with the pulse part, and therefore the new statistic Yp is introduced instead of TN. 1 $ = X$(F) s

dF

(1)

0

G(F)= G(F)=0

l/p

for for

1 -p
1, -p

if1

fLYI

a

1

L___________

7-

7

x [X(F)1

b

Figure

276

(1) The characteristics can be calculated by any data which have been obtained in the past. They should also be calculated as simply as the mean and the variance. (2) The properties of the data should be represented without distortion by those characteristics. (3) The maximum value i;;p should be calculated as a real number in any case. If the characteristics are unsuitable, the maximum value of Fp is evaluated as an imaginary number and cannot be used.

f(X)

1

Distributions

Appl.

Math.

where p is approximately equal to 2/N in case of N % 1 (see Appendix 1). The relation between Fp and distribution function X is discussed-later in the paper using Figure 1. The relation between Yp and the probability of failure is also discussed later. Let yp be called the modified mean largest value. The unknown function X(F) is obtained by solving a variational problem of maximizing r, under the restriction that X(F) must have the same characteristics as the data. The next problem is the selection of the characteristics used to restrict the population distribution. The characteristics should satisfy the following conditions:

X(F)

The characteristics such as mean, variance, maximum, minimum, block mean, skewness, kurtosis, etc. . . . are habitual statistics. Among them, the mean and the variance satisfy conditions (l)-(3) and are also required in order to normalize the data, therefore they are used as the constraints in this case. On the other hand, the maximum, the

and f(x)

Modelling,

1980,

Vol 4, August

A model for structural safety using the extreme value approach: S. Baba et al.

minimum and the block mean effectivzly prevent the overestimation of the maximum value of YP,4 but they sometimes give an imaginary maximum value.6 Furthermore, they are quite variable for each measurement ,e because they are calculated from only one part of the data. Thus, they do not satisfy conditions (2) and (3). The skewness and the kurtosis are less variable than the maximum, etc. . . . , and they are advantageous in this point. However, as a result, the skewness and the kurtosis are contrary to condition (l), because the maximum value of FP cannot be calculated analytically when they are used as the constraints. Therefore the authors introduce new characteristics, which correspond approximately to the skewness and the kurtosis, and by which the maximum value of FP can be calculated analytically. The conditions that the mean X is 0 and the variance s$ is 1, are expressed in the forms of the following integrals, as X is a normalized quantity: 1

XdF=O

(2)

s 0

s

X’dF=

1

(3)

0

The conditions that the skewness is /3aand the kurtosis is p4 are similarly expressed as follows:

s

I ,.

J

XF’dF=B

(4b)

I

XF3dF=C

(4c)

s

(4d)

0

1

0

XF4dF=D

0

The characteristics A-C of equations (4a-c) are direct constraints which characterize the unknown function X(F), while the characteristic D of equation (4d) is dummy and is employed in an undisclosed fashion to force X(F) into a monotone increasing function at the interval 0
X3dF=&

0

X(F)=(1/2h,){$(F)-hI-h,F-h,F2

1

- XsF3 - h6F4)

r

X4dF=/3‘,

J

0

Since they are unsuitable characteristics as previously mentioned, the authors introduce the following characteristics as the substitutes for skewness and kurtosis, respectively : 1

s

Substituting equation (5) into equations (2), (4a-d), the following simultaneous equations for X1 and X3--h6 are set up by leaving X2 unknown: [MI {h} = {C> - 2A* @I

1

112

l/3

114

115

l/3

114

l/5

l/6

115

l/6

117

117

118

WI =

I

1

X(F - O.5)3 dF = /i, 0

vm

l/9

{X}T=ol

The skewness and the kurtosis are obtained by averaging X for F according to the weights X2 and X3, respectively. In the same way, we can imagine the weighted mean where X is averaged for F according to the weight (F - O..5)2 instead of the weight X2. The weight (F - O..5)3can also be substituted for the weight X3. Furthermore, other characteristics are similarly imagined by averaging X for F according to the weights (F - 0.5) and (F - 0.5)4. Since the weights (F - OLY)~, k = 1,2,3,4 are expressed as quartic equations for F, we recommend substituting these four characteristics by the following new characteristics A, B, C and D.

(6)

where :

X(F - O.5)2 dF = 0;

0

(5)

{CF=

x3

A4

(1 - (1 -

A,

b}

p)2)/2p

fl - (1 7g3)/3p

(1 - (1 -P)41/4P {E}T={O

A

B

(1 - (1 -P)W5P)

C Dj

{h} is solved as : {A)= [M]-‘(c)where : 5

2h,[M]-‘{E} -60 960

210 -3780

[Ml-’ = 5

c

XFdF=A

15876 sym

(4a)

Appl.

Math.

Modelling,

(7) -280 5376 -23520 35840

1980,

Vol 4, August

277

A model for structural

safety using the extreme value approach:

S. Baba et al.

b

The unknown multiplier hz is obtained by substituting equations (5) and (7) into equation (3) as: (1 -

1’2xz =(I,p -

{E)qM]-‘{E))“”

(8)

(c)TIM]-l{c))“z

The maximum value of FP, that is, FF, is expressed as follows by substituting (5) into (1) and by using equations (7) and (8): ?-; = (1 - {_@[M]-‘{E))“’ x (l/p - {cJTIM]-l(c))l’* Let F: be called the modified value.

Property

and meaning

+ (c}T[n,r]-‘{E}

maximum

(9)

mean largest

of the function

X(F)

The property of the function X(F), which is expressed in equation (5), is explained in this section. The function X(F) defined in equation (5) is composed of a pulse G(F) and smooth functions of fourth order for F, that is, X(F) is a combination of a pulse part and a tail part as shown in Figure 1. At the interval (1 - p) < F < 1 the pulse part is not flat, but it is a monotone increasing function with smooth variations when p is a small number between 0 and 0.1. The shape of the probability density distribution f(X) = l/(dX/dF) is shown in Figure lb. The shape of the distribution f(X) shown in Figure lb is different from the population probability density distribution for X, but it has the same mean, the same variance and the same characteristics A-C as the population. The distribution f(X) is characterized by its pulse part located independently at the most right of the distribution. The average value of the pulse part is yP as defined in equation (1). The derivation of the maximum value of yP is based on the variational principle as mentioned in the previous section. In other words, in the extreme state, the pulse part is translated in a positive direction to the right as far as possible as shown in Figure 2a. The characteristics of the distribution f(X) are always constant independent of the location of the pulse part, because the distribution f(X) is always controlled to have a constant mean, a constant variance and constant characteristics A-C as restricted by equation (5). The relationship between the maximum value of FP and the maximum probability of failure is explained below based on the property of f(X).

Relationship between the modified maximum mean largest values and the maximum probability of failure First, it should be noted that there is no relationship between the modified maximum mean largest value Ff and the maximum probability of failure as long as they are used independently. They can be related to each other only when the modified maximum mean largest values are applied both for load and resistance at the same time. The probability density distribution for load,fs(X), is shown in Figure 2a, and the probability density distribution for resistance,&(X), in Figure 2b following the description in the previous section. When the maximization process is applied to the distribution fs(X), the pulse part

278

Appl. Math. Modelling,

1980,

Vol 4, August

Figure 2

Maximization

and minimization

of pulse parts

whose average value is YPs translates in a positive direction and reaches the maximum position as shown in Figure 2~. Let the average value of the maximum position be y&. On the contrary, when the minimization process is applied to t&e distribution fR(X), the pulse part whose average value Y,, translates in a negative direction and reaches the minimum position as shown in Figye 2b. Let the average value of the minimum position be Y&. In Figure 2, the distribution before the maximization process or before the minimization process is shown by dotted lines and the extreme state after the processes is shown by full lines. Applying these statistics to the structural design problem, the upper bound of load, Smax, is calculated as,

(104 the lower bound of resistance, R,i,, -E

Rmin =R - Y~R

‘SR

is calculated

as: (lob)

They are used as the design load and the design resistance, respectively. We should remember that a load larger than S ,.,,* and a resistance smaller than Rmin are expected not to occur following the variational principle. In other words, an extreme state is realized only when both S,,, and R,i, are employed as the design values. The extreme state is shown in Figure 2c, where load S is the abscissa and resistance R is the ordinate. In Figure 2c the region under a straight line R = S implies structural failure. When both pulse parts are disposed as shown in Figure 2c, the whole volume surrounded by the pulse parts and by the structural failure boundary line is approximately pspR/2, as the pulse parts are almost equal to a uniform distribution. In other words, the probability of failure is: pf =

PSPRI*

(11)

if the structure is designed under the condition S,, = Rmin. We should take note of equation (9) now. Equation (9) can be considered as a function of p, and the variable p is included in the terms such as (l/p), (C}T and {C). The

A model for structural safety using the extreme value approach: S. Baba et al.

characteristics

of each term in equation

(9) are as follows:

(1 - {E]Q4]-‘{E:)“*

constant

(I/p - (C}T[M]-‘{C})1’2

monotone decreasing function (Figure 30)

{C>T]~l-lJ~I

monotone decreasing function (Figure 3b) and [M]-‘(E) > 0

Therefore the modified maximum mean iargest value Ff is a monotone decreasing function of p in the interval O
Centre (me of main truss

b

(1) The shaded portion in Figure 3 implies that the occurrence of any combination of p and F: is equi-probable. (2) The p- FT line is the lower boundary probable combinations ofp and Ff.

line covering all

(3) A premise that p is a constant and FF is a variable, is reversed. Let y: b_ea constant and p is a variable which is subordinate to Y:. (4) The region of p for a certain pg is 0 < p
fj[ _-

Al AT

_

A3

2 4472 1 5801 0.7126 4.7399

Figure 4

Truss bridge and influence lines

and p can be easily determined

as follows:

p = (2Pf)l’2

(12)

After determiningp, the modified maximum mean largest value p: is calculated by using p, the upper and lower bounds S,, and Rmin are calculated by using F& and F&, respectively, and the structural design is performed by using S,, and R,in. In case of the design of a tension or a compression member, the cross-sectional area A, is determined in order to satisfy the following inequality: S max
(13)

Numerical example in structural design As an example of structural design, we calculate the crosssectional areas for a lower chord (tension) member and an upper chord (compression) member of a Warren truss bridge as shown by thick lines in Figure 4a. The bridge is a two-lane highway bridge for one-way traffic, whose cross-section is shown in Figure 4b. The data for yield and buckling strengths of the steel and axial forces of the chord members caused by the traffic flow are required in order to determine the cross-sectional area. The details of these data are as follows: P

C The occurrence of any

YfQF4

C Figure 3

Maximum

probability

of failure

(1) Yield strength of steel: The yield strength data of SM4lB steel measured by the Society of Steel Construction of Japan (JSSC) are used.9 The quality of SM41B steel is almost equivalent to that of A36 steel corresponding to ASTM Designation. (2) Buckling strength of steel: The buckling strength data of SS41 steel measured by Aoki and Fukumoto3 at the Department of Civil Engineering, Nagoya University are used. The test was performed for a welded H-column with a slenderness ratio of 100. The quality of SS41 steel is equivalent to SM41B steel.

Appl.

Math. Modelling,

1980,

Vol 4, August

279

A model for structural

safety using the extreme value approach:

S. Baba et al.

(3) Axial force of the chord member: The axle load record measured by the Japan Highway Public Corporation is used.1° The traffic flow model acting on the truss is substituted by the random series of the axial loads generated by the computer on the basis of the measured records covering over 3 1 weeks.

x 0.9 = 1.33 (t/m) with reference to Figure 4b by taking account of 10% reduction in the load intensity, and the total live loads acting on the chord members are:

The maximum stresses occurring in both members per week are used as a set of samples of load S. We use the maximum stresses per week as the load. When the structure is judged once a week on whether it fails or not, approximately 5000 judgements will be repeated during 100 years. Now, the structural life is 100 years, and let the structure be designed for the situation in which the structure will fail once during 100 years. In this case, the probability of failure is Pf = l/5000, that is, the area of the pulse part is p + l/SO according to equation (12). In general we are accustomed to using a smaller probability such as Pf = 10m6for the structural design. However, we should not forget that the maximum stress per week is used as the load in this example, and the situation will be changed if the maximum stress per hour or per minute is used as the load. The authors considered that this difference caused by the selection of time interval of sampling should be reflected in the decision on the probability of failure. The characteristics obtained from the data are given in Table 1. The modified maximum mean largest values FF are calculated for p = l/50 and are shown in Table I. Since they are represented as normalized values, the original values are obtained as follows with reference to Table 1:

and the required cross-sectional follows:

Yield strength Axial force

2.752 - 3.52 x 0.119 =2.333 (t/cm*) 14.44t4.19 x 1.95=22.61(t)

Buckling strength Axial force

1.25 1 - 3.64 x 0.156 = 0.683 (t/cm*) 28.88+4.19x 3.91=45.22(t)

Therefore the cross-sectional are calculated as follows :

A, = 22.61/2.333

= lO(cm*)

Upper chord

A, = 45.22/0.683

= 67 (cm*)

Modified

maximum

16.5 x 0.875Ot1.33

x 35.00=61.0(t)

areas A, are calculated

Lower chord

A, = 30.5/l .40 = 22 (cm’)

Upper chord

A, = 6 1.0/0.9 1 = 68 (cm’)

as

Lower chord

Pf = 2 x 1O-6

Upper chord

Pf = 2 x 10e4

A new statistic 7: has been introduced with the aim of relating the maximum mean largest value concept to the maximum probability of failure. By calculating the upper bound of load and the lower bound of resistance, a sort of extreme state design has been realized. New characteristics A, B, C have been introduced in order to improve the overestimation of the maximum mean largest value. The extreme value approach described here is characterized by the following points: (1) The maximum mean largest value concept based on the variational principle has been applied first to the structural design problem. The design obtained as a result has been in an extreme state, where the probability of failure of the structure never exceeds an initially settled value P,-, for which the structure is designed. (2) This is a distribution-free approach to the concept of structural safety, and the probability of failure has been calculated without assuming the probability density distributions of load and resistance. (3) The characteristics A, B, C have been introduced SO that the character of the data has been strongly reflected

mean largest value of data Unit

”

x

sx

Yield strength (SM41B) Buckling strength (SS41) Axial force (lower chord) Axial force (upper chord)

t/cm2 t/cm’ t t

21 48 31 31

2.752 1.251 14.44 28.88

0.119 0.156 1.95 3.91

Appl.

Upper chord

17.50=30.5(t)

Conclusions

Measured data

280

x

It is presumed that the difference of the probabilities of failure between the lower chord and the upper chord is originated by the difference of the reliability level between the tension test9 and the buckling test,3 as the coefficient of variation of the buckling strength is almost three times greater than that of the yield strength, as shown in Table I. Although the extreme value approach is said to give excessively large extremes, the numerical example gives a reasonable result, which is not much removed from the result obtained by a conventional design method. Based on this example, the authors cannot assert that the extreme value approach would give a reasonable result in all cases; however, the authors believe that this approach has potential for the solution of various technical problems.

We can estimate the maximum probability of failure of the same chord members, when they are designed in accordance with the AASHTO code for the highway bridges. With respect to resistance, the allowable stresses for the tension member and the compression member are, respectively, 20 000 (psi) + 1.40 (t/cm2) and 13 000 (psi) + 0.9 1 (t/cm*), since A36 steel is employed. With respect to load, the concentrated load is 26 000 (lb/lane) + 3.87 (t/m) and the uniform load is 640 (lb/ft-lane) f 0.3 12 (t/m*) if H-20 lane loading is adopted. The relevant influence lines are shown in Figure 4 for the main truss and for the slab. The lane loadings are placed on the slabs as shown in Figure 4b in order to maximize the stresses of the main truss. The concentrated load and the uniform load on the slab are, respectively, 3.87 x 4.740 x 0.9 = 16.5 (t) and 0.312 x 4.740

Table 7

16.5 x 0.4375 + 1.33

The maximum probability of failure corresponding to the above cross-sectional areas are approximately :

areas A, of the chord members

Lower chord

Lower chord

Math. Modelling,

1980,

Vol 4, August

6,

0.043 0.124 0.135 0.135

A”

0.2577 0.2758 0.2663 0.2663

E

0.2467 0.2829 0.2736 0.2736

(7

0.2142 0.2608 0.2528 0.2528

iq

3.52 3.64 4.19 4.19

A model for structural safety using the extreme value approach: S. Baba et al. on the estimation of the maximum mean largest value. The characteristics have been stable, that is, they have remained almost unchanged even for a slight variation in the data.

sample variances of R and S normalized value of x, that is, X = (x - x)/s, probability density for F, the unknown function order statistics of X random variables (data) mean largest value modified mean largest value modified maximum mean largest value modified maximum mean largest values for R and S modified minimum mean smallest value skewness and kurtosis characteristics corresponding approximately to fi3 and &, coefficient of variation Lagrange multipliers pulse function which is $(F) = I/p for

(4) The dummy characteristic D has been introduced to the definition of the modified maximum mean largest value in order to force the unknown function X(F) into a monotone increasing function. (5) A numerical example of the extreme state design has been provided by employing the test data, and a reasonable result has been obtained compared with the allowable stress design. The approach is believed to have potential in the solution of various technical problems.

Acknowledgements The data used here have been supplied by Dr Aoki, the Society of Steel Construction of Japan and the Japan Highway Public Corporation. The authors also wish to thank Professor Massonnet and Dr Dotreppe of the University of Liege for their assistance.

References 1 2 3

8 9 10

11 12

I-p
Appendix

Ang, A. H.-S. and Amin, M. Proc. ASCE, 1968,94, (EM2), 559 Ang, A. H.-S. and Ellingwood, B. R. In Proc. 1st Int. Conf Appl. Stat. Probability Soil Struct. Engng, Hong Kong, 1971 Aoki, T. and Fukumoto, Y. Proc. JSCE, 1974,222, 37 (in Japanese) Baba, S. etal. Trans. AIJ, 1975,233, 13 (in Japanese) Baba, S. etal. Proc. JSCE, 1977,267,27 (in Japanese) Baba, S. Dr. Thesis, Nagoya University, 1977 Gumbel, E. I. Ann. Inst. Henri Poincare, 1935,4,115 (in French) Gumbel, E. J. ‘Statistics of extremes’, Columbia University Press, 1950 Horikawa, K. Proc. JSSC, 1969,5, 52 (in Japanese) Japan Highway Public Corporation. Measuring records of traffic volume census at Tennozan Tunnel, Meishin Highway (in Japanese), 1968 Plackett, R. L.Biometrika, 1947, 34, 120 Freudenthal, A. M. Trans. ASCE, 1947,112,125

1

Relation between

p,” and Yg

The maximum of i;p” is already obtained by equation (9), whereas the maximum of ?i is represented as follows: ?N” = (1 - {E)T[M]-1~E})“2(N2/(2N

- 1)

- {C*]T[M]-’ (c*))“2 •t (c*J[M] -‘(El {C*jr=

(1 N/(N+

There are two differences as follows : (1) difference between (2) difference between

1) N/(N+2)

N/(N+3)

between equations

(Al) N/(N+4))

(9) and (Al)

{C} and {C*] I/p and N2/(2N - 1)

If we assume the following relationship

between p and N:

p = 2/N

Notation

(A2)

the differences can be neglected as follows in case of N % 1:

D F(X) 5 rrx”f fR,fS &‘A

J N n pf P PR>PS

R R Rmin S s S ““: sx

characteristics of data means of characteristics A, B, C cross-sectional area of chord member of truss bridge dummy characteristic cumulative function of f(X) ith value of F mean of Fi probability density for X probability densities for R and S probability density for Fi = F(Xi) functional size of sample for which maximum mean largest value is defined size of the data probability of failure area of the pulse part areas of the pulse parts for R and S resistance (material strength) sample mean of R lower bound of R load (external force) sample mean of S upper bound of S variance of data

[iC*) - {C)l/iC*I [l/p - N2/(2N

= (W/N2)1

- l)]/[N’/(ZN

- l)] = O( l/N)

Although there is no physical relation between $ and yi, they can be regarded as equivalent in the limited case of p = 2/NandN% 1.

Appendix

2

Calculation

of the characteristics A, B and C

The characteristics A, B, C defined by equations (4a-4c) are -estimated in the form of ‘means of A, B, C’, that is, A, B and C, based on the order statistics X1
ZXiFi

A, 2, is then expressed as:

= (l/n) CXi&

Further, the characteristic mean dis expressed more simply as follows, as the maximum likelihood estimator of zi is Xi: .Z= (I/n) ZXiFi

Appl.

Math.

(A3 a)

Modelling,

1980,

Vol 4, August

281

A model for structural

The characteristic follows :

safety using the extreme value approach:

Appendix

means B and C?are also expressed as

Calculation

B = (l/t') CXiFf

Wb) Since the probability as:8

density g(Fi) for Fi = F(Xi) is defined

g(Fi) = (TZ!/(~ __ l)! (i - l)!} F!-‘(1 the statistics Fi, Ff, F: in equations follows :

-Fi)n-i,

(A3) are expressed as

F;

= i(i t l)/(n + l)(n + 2)

F;

= i(i + l)(i + 2)/(n + l)(n + 2)(n + 3) the characteristic

mean 2, B, care

defined

Ci(i + 1) Xi

(A4)

C= {l/VZ(fit l)(n t 2)(n + 3)] Ci(i •t I)(i + 2) Xi The modified minimum mean smallest value yp”’ is calculated directly from equation (7) if the characteristics A, B, C are redefined as follows: (1) The order statisticsXr!-X,_i+r (X;
282

Appl. Math. Modelling, 1980, Vol 4, August

of dummy characteristic D

The dummy characteristic

D defined by equation (4d) is determined adequately in order to force X(F) into a monotone increasing function. Since X’(F) should always be positive, the following inequality is obtained from equation (5):

in which the coefficient of G’(F), that is, h2 should always be positive, because G’(F) corresponds to Dirac’s distribution 6(F - 1 + p); the other coefficients h3-h6 should be positive as a whole. Therefore the following simultaneous inequalities are obtained:

h3 t 2X4F + 3X5F2 + 4h6F3 < 0

a= (l/?Z(n + 1)) CiXi B = (l/iZ(?I + l)(n + 2))

3

(1/2Xz)(*‘(F)-X3-2X4F-3hgF2-44XsF3)>0

Fi = g(Fi) Fi dFi = i/(n + 1)

In conclusion, as follows :

S. Baba et al.

The range of D is determined by solving equation (A5) with respect to D in the interval 0 0,

x3<0,

xs>o,

-3xg
;\4 < - (3 A5 •t As)/2

Next these equations are expressed as simultaneous inequalities of high degrees for D, because h2-h6 are directly represented by D through equations (7) and (8). The range of the dummy chiracteristic D is then calculated by solving the simultaneous inequalities for D (see reference (6) for the detailed explanation). The modified maximum mean largest value y‘ is not constant in the effective range of the dummy characteristic D. We recommend usin the smallest value of Ff, -9. 1s equally safe as expressed because any selection of Y, in the previous section with regard to ‘How should we determine ps and pR . . . ?‘.