MSc_SA&L_L03_ReliabilityNotes_2008_2009[1]

8/8/2019 MSc_SA&L_L03_ReliabilityNotes_2008_2009[1]

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Fundamentals of Structural Reliability Analysis

Dr Peter J. Stafford1

February 2009

1RCUK Fellow / Lecturer in Modelling Engineering Risk; Willis Research Fellow; Room 405; [email protected]


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Contents

1 Limit-state Functions and the Reliability Problem 2

2 First Order Reliability Method (FORM) 6

2.1 Cornell Reliability Index . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Hasofer and Lind Reliability Index . . . . . . . . . . . . . . . . . . . 9

2.2.1 Linear limit-state function . . . . . . . . . . . . . . . . . . . . 10

2.2.2 Sensitivity Factors . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Interpretation of First-Order Second-Moment (FOSM) Theory . . . 12

A Rosenblatt Transformation 13

List of Figures

1 Visual representation of the failure domain Fand the limit-statesurface G = 0 for the basic reliability problem. . . . . . . . . . . . . 3

2 Graphical interpretation of the two specifications of the convolu-

tion integrals given in equations 8 (top) and 9 (bottom). . . . . . . . 3

3 Probability density function of the safety margin, G. Also shown isa graphical interpretation of the reliability index, . The shaded area

represents the probability of failure and is equal to the area under

the PDF from 0. . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Various demonstrations of how the reliability relates to normally

distributed limit-state functions. The top figure shows how the

probability of failure is defined for the original distribution; the

middle figure shows the corresponding probability of failure for a

distribution whose mean has been shifted to have a value of zero;and the bottom figure shows a limit-state distribution transformed

so that the reliability index now represents a direct distance from

the origin. In all cases, the shaded area represents the same proba-

bility of failure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

5 Cantilever beam loaded by two point forces . . . . . . . . . . . . . . 7

6 Graphical representation of limit-state surfaces and their equiva-

lence at g(r,s)=0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

7 Limit-state surface G(x) = 0 and its linearised version GL(x) = 0in the space of original basic variables; the X-space . . . . . . . . . 10

8 Probability density function contours and original (non-linear) and

linearised limit-state surfaces in the standard normal space. . . . . 11

9 Marginal distribution in the space of standardised normal vari-

ables. The marginal distribution correponds to the axis drawn

through points O and P in figure 8. . . . . . . . . . . . . . . . . . . . 11

10 Inconsistency between and Pf for different forms of limit state

functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

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S

R

G=0 o

rR

=S

G < 0 or R < S

Figure 1: Visual representation of the failure domain

Fand the limit-state surface

G = 0 for the basic reliability problem.

Now recalling the definition of the cumulative distribution function2 one may

appreciate that in both of the above cases the inner integral may be written in

terms of either the CDF ofR (equation 8) or the CCDF of S(equation 9). In doing

so one obtains what is known as the convolution integral.

Pf =s=

sr= f

R(r)dr

fS(s)ds = F

R(s)fS(s)ds (8)

Pf =

r=

s=r

fS(s)ds

fR(r)dr =

[1 FS(r)] fR(r)dr (9)

Note that the product FR(r)fS(s) is known as the failure density. It may not be

immediately obvious what the meaning of the convolution integrals in equations

8 and 9 is, and more importantly, why this representation of the integral may be

more preferable. However, these equations may be interpreted easily by consid-ering figure 2. In the top half of figure 2 the first convolution integral is depicted

(equation 8). This figure may be interpreted as follows. The total probability of

failure is made up of the sum of the probabilities corresponding to all cases where

the resistance R is less than the loading S. The probability that the loading takes

on a particular value is approximately P(S = s) = fS(s)ds, this probability isshown by the heavy shaded area in the figure. The probability that the resistance

is less than or equal to this level of loading is given by the CDF of the resistance

2FX(x) =x

fX()d - note also that the complementary cumulative distribution function

(CCDF) is defined as FX(x) = 1 FX(x).

evaluated at this level of loading, i.e., FR(s). The product of these two terms rep-

resents the probability that this particular level of loading causes failure. The total

probability is then found by considering all such scenarios and is represented by

the integral over all possible levels of loading, s, stipulated in equation 8.

R or S

fR(r) or fS(s)

R or S

fR(r) or fS(s)

fS(s) fR(r)

ds

fS(s)ds

FR(s)

fS(s)

dr

fR(r)

fR(r)dr

1 - FS(r)

Figure 2: Graphical interpretation of the two specifications of the convolution in-

tegrals given in equations 8 (top) and 9 (bottom).

Likewise, in the bottom half of figure 2 the second convolution integral is graph-

ically represented (equation 9). In a similar vein to the previous discussion, this

figure may be interpreted as follows. The total probability of failure is made upof the sum of the probabilities corresponding to all cases where the loading S is

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0 G G

fG(g)

- G 0 G = G- G

fG (g)

- 0 G*= (G- G)/ G

fG*(g*)= (g*)

G N(G

,G

2)

Pf = FG(g=0| G, G)

G

Pf = FG (g=- G|0, G)N(0, G

2)

N(0,1)Pf = (g*=- )

Figure 4: Various demonstrations of how the reliability relates to normally dis-

tributed limit-state functions. The top figure shows how the probability of failure

is defined for the original distribution; the middle figure shows the correspond-

ing probability of failure for a distribution whose mean has been shifted to have a

value of zero; and the bottom figure shows a limit-state distribution transformed

so that the reliability index now represents a direct distance from the origin. In allcases, the shaded area represents the same probability of failure.

previously encountered, i.e., ln R ln S > 0. As both R and Sare lognormally dis-tributed, ln R and ln S are both normally distributed; we may therefore follow a

very similar procedure to that adopted for normally distributed random variables

with the only exception being that we must transform the first moments of the

logarithmically distributed variables R and S such that they are appropriate for

the problem in question. The variances ofln R and ln Sare found using equations

12 and 13,

2lnR = ln

1 +

RR

2= ln

1 + V2R

(12)

2lnS = ln

1 +

SS

2= ln

1 + V2S

(13)

where VR and VS are the coefficients of variation of R and S respectively. Themeans of the logarithms ofR and S are functions of the above variances and may

be found from equations 14 and 15.

lnR = ln R 12

2lnR (14)

lnS = ln S 12

2lnS (15)

With these expressions defined we may now directly determine the probabilityof failure using the CDF of the standard normal distribution with the arguments

appropriate for normally distributed random variables:

Pf =

(lnR lnS)

2lnR + 2lnS

(16)

Consider now the relationship between the reliability formulation just presented

and the more traditional factor of safety methods that were discussed in the first

lecture. One can define the most common factor of safety as the ratio of the meanvalues of the resistance and the loading. This factor of safety is known as the

central safety factor and is written:

0 =RS

(17)

we are all familiar with this expression and know that values of 0 greater than

unity correspond to safe states while values less than unity correspond to unsafe

states. However, we also know that generally when dealing with code specifica-tions we do not use factors that relate directly to the expected values of material

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11.5

22.5

3

11.522.5

3-10

-5

0

5

10

resistan

ce,R

loading, S

lim

it-statefunction,g(r,s)

g(r,s) = r2 - s2

g(r,s) = ln(r/s)g(r,s) = r- s

g(r,s) = 0

Figure 6: Graphical representation of limit-state surfaces and their equivalence at

g(r,s)=0

from the limit-state surface. The reliability indices calculated previously are de-

termined for the expected values of both R and S and the Taylor series approxi-

mation results in tangent planes to the three surfaces being used to determine the

reliability. However, these tangent planes, when evaluated away from the limit-

state surface, do not intersect the r

s plane at the same point as the real nonlinear

surfaces with the result being different values of the reliability index [4].

In other words, when one interprets the reliability index as being a distance

away from the expected values of the random variables, the distance determined

from where the tangent plane intersects the failure surface is different to the ac-

tual distance that corresponds to the intersection of the nonlinear surface with the

failure surface. The combination of the approximation via linearisation and the

expansion about the expected values thus results in the differences between the

calculated C values.

If the reliability index is supposed to be directly related to the probability of fail-

ure of a structural component or system it is obviously troublesome that the index

may differ significantly when using the Cornell approach for nonlinear limit-state

functions. This problem is known as invariance and has a solution that we will

discuss in the following section.

2.2 Hasofer and Lind Reliability Index

In order to overcome the lack of invariance of the Cornell index, Hasofer and

Lind [3] proposed a different definition for the index which coincides with the

Cornell index when the limit-state function is linear, but possesses the property

of invariance with respect to any form of nonlinear limit-state function [6]. The

essence of the Hasofer and Lind [3] approach consists of transforming the random

variables involved in the problem to obtain a set of normalized and uncorrelated

variables. In the case of problems involving uncorrelated random variables this

transformation is very straightforward, one simply maps variables inX

intoY

using the following formula that we have already encountered:

Yi =Xi Xi

Xi(44)

the resulting vector of the expected values ofY is now simply Y = 0 and the co-

variance matrix ofY is just the identity matrixCY = I. We have previously talked

of interpreting the reliability index as a distance measured in units of standard

deviations from the mean value of a random variable. Continuing with this in-terpretation, one may recognise that whereas in the X-space the unit of measure

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Figure 8: Probability density function contours and original (non-linear) and lin-

earised limit-state surfaces in the standard normal space.

By well known properties of the bivariate normal distribution the marginal dis-

tribution is also normal, and hence the shaded area in figure 9 represents the fail-ure probability Pf = (), where is as shown (note that = 1 in the di-rection since the normalised Y-space is being used). The distance showin in

figure 9 is perpendicular to the axis and hence is perpendicular to g(y) = 0. It

clearly correponds to the shortest distance from the origin in the Y-space to the

limit-state surface g(y) = 0.

More generally, there will be many basic random variables X = {Xi, i =1, . . . , n} describing the structural reliability problem. In the case of complexstructures, n could be very large indeed. Evidently this will create a problem forintegration methods. However, this curse of dimensionality is not so critical for the

First Order Second Moment method since the concepts described above carry di-

rectly over to an n-dimensional standardised normal space y with a (hyper)plane

limit-state. In this case the shortest distance and hence the reliability index is given

by:

= min

ni=1 y

2i

= min

yTy

subject to g(y) = 0(48)

where the yi represent the co-ordinates of any point on the transformed limit-state

0

fY(y)

Pf

g(y) = 0

Figure 9: Marginal distribution in the space of standardised normal variables.

The marginal distribution correponds to the axis drawn through points O and P

in figure 8.

surface. The particular point that satisfies equation 48, i.e. the point on the limit-

state surface perpendicular to , in n-dimensional space, is known as the design

point y. Evidently this point is the projection of the origin on the limit-state sur-

face. It should be obvious from figures 8 and 9 that the greatest contribution to

the total probability content in the failure region is that made by the zone close to

y. In fact, y represents the point of greatest probability density or the point of

maximum likelihood for the failure domain. A direct relationship between the de-sign point y and can be established as follows. From the geometry of surfaces

the outward normal vector to a hyperplane given by g(y) = 0, has components

given by:

ci = g

yi(49)

where is an arbitrary constant. The total length of the outward normal is

l =

i

c2i (50)

and the direction cosines i of the unit outward normal are then

i =cil

(51)

With i known it follows that the co-ordinates of the design point are

yi = yi = i (52)

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Figure 10: Inconsistency between and Pf for different forms of limit state func-

tions

independent variables:

n(y) =ni=1

12

exp1

2y2i

(58)

The reliability index associated with a given limit state (hyper-)surface with

safe domain Sk, say, is then obtained by integrating n(y) over the domain Sk togive the value for the safe state as:

[(k)] = Sk

n(y)dy (59)

where () is the standardised normal distribution function. It follows that

(k) = 1

Sk

n(y)dy

(60)

provides the definition of the general reliability index. It is clearly a function

of the shape of the limit state function for the safe domain Sk. It is probably alsoreadily appreciated that the solution to the above equation may be rather complex

owing to the potentially complex shape ofSk.

A Rosenblatt Transformation

We have seen that common reliability problems are most simple when dealing

with vectors of independent random variables. Fortunately, there exist techniques

via which vectors of dependent random variables may be transformed into a cor-

responding vector of independent random variables. This appendix deals witharguably the most common such transformation, the Rosenblatt transformation [7].

A dependent random vector X = {X1, X2, , Xn} may be transformed into theindependent uniformly distributed random vectorR = {R1, R2, , Rn} throughthe Rosenblatt transformation [7] R = TX defined by:

r1 = P (X1 x1) = F1(x1)r2 = P (X2 x2|X1 = x1) = F2(x2|x1)...

rn = P(Xn xn|X1 = x1, . . . , X n1 = xn1) = Fn(xn|x1, . . . , xn1)(61)

In the representation of equation 61 the Fi() is shorthand for the cumulative con-

ditional distribution function that is more formally given by FXi|Xi1,...,X1(). If

the joint probability density function fX() is known then Fi() can be determined

as follows. The conditional probability density function fi() is given by

fi(xi|x1, . . . , xn1) =fX

i

(x1, . . . , xi)

fXi1(x1, . . . , xi1)(62)

where fXj (x1, . . . , xj) is a marginal probability density function obtained from

fXj (x1, . . . , xj) =

. . .

fX(x1, . . . , xn)dxj+1, . . . , d xn (63)

Fi() is then obtained by integrating fi() given in Equation 62 over xi:

Fi(xi|x1, . . . , xn1) = fXi(x1, . . . , xi1, t)dt

fXi1(x1, . . . , xi1)(64)

With all of the conditional cumulative distribution functions Fi() determined in

this way, equation 61 may be inverted successively to obtain

x1 = F11 (r1)

x2 = F12 (r2|x1)

.

..xn = F

1n (rn|x1, . . . , xn1)

(65)

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It follows immediately that equation 65 can be used to generate the random vector

X with probability density function fX() from R.

Note that the indexing used when writing out Equation 61 is arbitrary and that,

as noted by Rosenblatt [7], there are n! possible ways in which the expressions

in equation 61 may be written, depending upon the numbering adopted for the

variables inX. For the same reason there are also n! possible ways of conditioningthe Xi in equation 61. This may be demonstrated for the relatively trivial case of

n = 2 [8].

FX1X2(x1, x2) = FX1(x1)fX2|X1(x2|x1) = FX2(x2)fX1|X2(x1|x2) (66)

It may be appreciated that there will be particular orders of operations that will

lead to greater ease in solving for X, i.e. in solving equation 65.

The Rosenblatt transformation may be used to transform from one distributioninto another by applying equation 61 twice, using R as a transmitter, e.g.

F1(u1) = r1 = F1(x1)

F2(u2|u1) = r2 = F2(x2|x1)...

Fn(un|u1, . . . , un1) = rn = Fn(xn|x1, . . . , xn1)

(67)

A particular case of interest in where U in equation 67 is standard normal dis-tributed, with X, say, a vector of correlated random variables and U uncorrelated

(independent). Then equation 67 may be written as

x1 = F11 [(u1)]

x2 = F12 [(u2)|x1]

...

xn = F1n [(un)|x1, . . . , xn1]

(68)

Note that in practice the solution of equation 68 requires multiple integration.

References

[1] C. A. Cornell. A probability based structural code. Journal of the American

Concrete Institute, 66(12):974985, 1969.

[2] O. Ditlevsen. Generalized second moment reliability index. Journal of Struc-tural Mechanics, 7(4):435451, 1979.

[3] A. M. Hasofer and N. C. Lind. Exact and invariant second moment code for-

mat. Journal of the Engineering Mechanics Division, ASCE, 100:111121, 1974.

[4] H. O. Madsen, S. Krenk, and N. C. Lind. Methods of Structural Safety. Prentice-

Hall international series in civil engineering and engineering mechanics.

Prentice-Hall Inc., 1986.

[5] R. E. Melchers. Structural reliability analysis and prediction. John Wiley & Sons

Ltd., Chichester, 1999.

[6] P. E. Pinto, R. Giannini, and P. Franchin. Seismic Reliability Analysis of Struc-

tures. IUSS Press, 2004.

[7] M. Rosenblatt. Remarks on a multivariate transformation. The Annals of Math-

ematical Statistics, 23:470472, 1952.

[8] R. Y. Rubinstein. Simulation and the Monte Carlo Method. John Wiley & Sons

Ltd., New York, 1981.

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MSc_SA&L_L03_ReliabilityNotes_2008_2009[1]