structural reliability and risk analysis courses

7/27/2019 Structural Reliability and Risk Analysis Courses

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1. INTRODUCTION TO RANDOM VARIABLES THEORY................................................ 31.1 Data samples .................................................................................................................... 41.2 Indicators of the sample (esantion) .................................................................................. 41.3 Probability ........................................................................................................................ 61.4 Random variables .............................................................................................................7

1.5 Indicators of the probability distributions ........................................................................ 92. DISTRIBUTIONS OF PROBABILITY.............................................................................. 12

2.1. Normal distribution ....................................................................................................... 122.2. Log-normal distribution ................................................................................................ 162.3. Extreme value distributions........................................................................................... 18

2.3.1. Gumbel distribution for maxima in 1 year ............................................................. 192.3.2. Gumbel distribution for maxima in N years........................................................... 22

2.4. Mean recurrence interval............................................................................................... 243. FUNCTION OF RANDOM VARIABLES ......................................................................... 26

3.1 Second order moment models ........................................................................................ 264. STRUCTURAL RELIABILITY ANALYSIS..................................................................... 30

4.1. The basic reliability problem......................................................................................... 304.2. Special case: normal random variables ......................................................................... 324.3. Special case: log-normal random variables................................................................... 334.4. Calibration of partial safety coefficients ....................................................................... 34

5. SEISMIC HAZARD ANALYSIS........................................................................................ 385.1. Deterministic seismic hazard analysis (DSHA) ............................................................ 385.2. Probabilistic seismic hazard analysis (PSHA) .............................................................. 395.3. Earthquake source characterization............................................................................... 405.4. Predictive relationships (attenuation relations) ............................................................. 435.5. Temporal uncertainty .................................................................................................... 435.6. Probability computations............................................................................................... 445.7. Probabilistic seismic hazard assessment for Bucharest from Vrancea seismic source . 45

6. INTRODUCTION TO RANDOM (STOCHASTIC) PROCESSES THEORY.................. 516.1. Background ................................................................................................................... 516.2. Average properties for describing internal structure of a random process ................... 526.3. Main simplifying assumptions ...................................................................................... 556.4. Other practical considerations ....................................................................................... 60

7. POWER SPECTRAL DENSITY OF STATIONARY RANDOM FUNCTIONS.............. 617.1. Background and definitions .......................................................................................... 617.2. Properties of first and second time derivatives ............................................................. 647.3. Frequency content indicators ........................................................................................ 64

7.4. Wide-band and narrow-band random process............................................................... 667.4.1. Wide-band processes. White noise......................................................................... 667.4.2. Narrow band processes........................................................................................... 68

7.5. Note on the values of frequency content indicators ...................................................... 70

References:

1. Structural reliability and risk analysis, Lecture notes(prof.dr.ing. Radu Vacareanu)http://www.utcb-ccba.ro/attachments/116_Structural%20Reliability%20Lecture%20Notes.pdf

2. Siguranta constructiilor (Dan Lungu & Dan Ghiocel)


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STRUCTURAL RELIABILITY AND RISK ANALYSIS4th

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Examination subjects:

1. Random variables, indicators of the sample, histograms2. Probability, probability distributions function, cumulative density function, indicators

3. Normal distribution4. Log-normal distribution5. Extreme value distributions, Gumbel distribution for maxima in 1 year and in N years,Mean recurrence interval6. Function of random variables and second order moment models7. Structural reliability analysis. Special cases8. Calibration of partial safety coefficients9. Seismic hazard analysis10. Random (stochastic) processes theory (average properties of a process)11. The simplifying assumptions used in random processes theory12. Definitions of power spectral density of stationary random functions

13. Frequency content indicators


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1. INTRODUCTION TO RANDOM VARIABLES THEORY

Reliability the ability of a system or component to perform its required functions understated conditions under a specified period of time.

Structural reliability objective to develop criteria and methods aiming at ensuring thatbuildings (structures) built according to specifications will not fail to preserve the functionsfor a specified period of time.

Uncertainty

- Random (aleatory) uncertainties that are related to the built in reliabilityphenomena and informations that cannot be reduced through better knowledge.

- Epistemic uncertainties that are related to our inability to predict the futurebehavior of systems. It can be reduced through better understanding and modeling.

Example: Analysis of a frame structure

Steps: Model (1) Imposed Loads (2) Structural Analysis (3) Sectional Analysis (4) Design for some limits states (5).

Model (1) - Epistemic uncertaintyLoads (2)dead, wind, live loads are evaluated using the codes.Random uncertainty (in the codes) - one way to take care, is the use of the safety factors (5%less than or 5% greater thansee the class of concrete)Structural Analysis (3) - Epistemic uncertainty (linear elastic materialsstrains proportionalto stresses)Sectional Analysis (4) -Random + Epistemic uncertainty

Design for some limits states (5) ultimate limit states (safety), serviceability limit states.For power plantsLOCAlocal use of cooling agent

Uncertainties can be reduced to physical models:- random variable (RV)dead load, material strength- random processwind velocity, ground acceleration.

The random variable (RV) is a quantity whose value cannot be predicted with sufficientaccuracy before performing the experiment.

In engineering statistics one is concerned with methods for designing and evaluating

experiments to obtain information about practical problems, for example, the inspection ofquality of materials and products. The reason for the differences in the quality of products is

Live load + dead loads

A

N M

Compressiveforce

Bendingmoment

As

As

Lateral load(wind,earthquake)

Steel area


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the variationdue to numerous factors (in the material, workmanship) whose influence cannotbe predicted, so that the variation must be regarded as a random variation.

1.1 Data samples

In most cases the inspection of each item of the production is prohibitively expensive andtime-consuming. Hence instead of inspecting all the items just a few of them (a sample) areinspected and from this inspection conclusions can be drawn about the totality (the

population).

If X = random variable(concrete compressive strength)x = value for random variables

If one performs a statistical experiment one usually obtains a sequence of observations. Atypical example is shown in Table 1.1. These data were obtained by making standard tests forconcrete compressive strength. We thus have a sample consisting of 30 sample values, so that

the size of the sample is n=30.

1.2 Indicators of the sample (esantion)

One may compute measures for certain properties of the sample, such as the average size ofthe sample values, the spread of the sample values, etc.

The mean value of a sample x1, x2, , xnor, briefly, sample mean, is denoted by_

x (or mx) andis defined by the formula:

n

j

jxn

x1

_ 1(1.1)

It is the sum of all the sample values divided by the size n of the sample. Obviously, it

measures the average size of the sample values, and sometimes the term averageis used for_

x .

The variance (dispersie) of a sample x1, x2, , xnor, briefly, sample variance, is denoted bysx

2and is defined by the formula:

n

j

jx xxn

s1

2_

2 )(1

1(1.2)

Thesample varianceis the sum of the squares of the deviations of the sample values from the

mean_

x , divide by n-1. It measures the spread or dispersion of the sample values and isalways positive.

The square root of the sample variance s2is called the standard deviationof the sample and is

denoted by sx.2

xx ss . The mean, mxand the standard deviation, sxhas the same units.

The coefficient of variation of a sample x1, x2, ,xnis denoted by COVand is defined as theratio of the standard deviation of the sample to the sample mean

x

sCOV (dimensionless) (1.3)


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Table 1.1. Sample of 30 values of the compressive strength of concrete, daN/cm2

320 380 340 360 330 360 380 360 320 350350 340 350 360 370 350 350 420 360 340370 390 370 370 400 360 400 350 360 390

The statistical relevance of the information contained in Table 1.1 can be revealed if one shallorder the datain ascending order in Table 1.2, first column (320, 330 and so on).

Table 1.2 Frequencies of values of random variable listed in Table 1.1

Compressivestrength

Absolutefrequency

Relativefrequency

Cumulativefrequency

Cumulative relativefrequency

320 2 0.067 2 0.067330 1 0.033 3 0.100340 3 0.100 6 0.200350 6 0.200 12 0.400

360 7 0.233 19 0.633370 4 0.133 23 0.767380 2 0.067 25 0.833390 2 0.067 27 0.900400 2 0.067 29 0.967410 0 0.000 29 0.967420 1 0.033 30 1.000

The number of occurring figures from Table 1.1 is listed in the second column of Table 1.2. Itindicates how often the corresponding value x occurs in the sample and is called absolute

frequency of that valuex in the sample.

Dividing it by the size n of the sample one obtains the relative frequency listed in the thirdcolumn of Table 1.2.

If for a certain value x one sums all the absolute frequencies corresponding to the samplevalues which are smaller than or equal to that x, one obtains the cumulative frequencycorresponding to thatx. This yields the values listed in column 4 of Table 1.2. Division by thesize n of the sample yields the cumulative relative frequency in column 5 of Table 1.2.

The graphical representation of the sample values is given by histograms of relativefrequencies and/or of cumulative relative frequencies (Figure 1.1 and Figure 1.2).

Figure 1.1 Histogram of relative

frequencies

Figure 1.2 Histogram of cumulative

relative frequencies


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If a certain numerical value does not occur in the sample, its frequency is 0. If all the n valuesof the sample are numerically equal, then this number has the frequency n and the relativefrequency is 1. Since these are the two extreme possible cases, one has:

- The relative frequency is at least equal to 0 and at most equal to 1;

- The sum of all relative frequencies in a sample equals 1.

If a sample consists of too many numerically different sample values, the process ofgroupingmay simplify the tabular and graphical representations, as follows (Kreyszig, 1979).

A sample being given, one chooses an interval I that contains all the sample values. Onesubdivides I into subintervals, which are called class intervals. The midpoints of thesesubintervals are calledclass midpoints.The sample values in each such subinterval are said toform a class.The number of sample values in each such subinterval is called the corresponding classfrequency (absolute frequency - nj).Division by the sample size n gives the relative class frequency (relative frequency -

n

nf

j

j and

m

j

jf1

= 1). The normalized relative frequency isx

ff

jN

j . The normalization is

with respect to thex.

The relative frequency is called the frequency function of the grouped sample, and thecorresponding cumulative relative class frequencyis called the distribution function of the

grouped sample (Fj=

j

kjf

1

).

If one chooses few classes, the distribution of the grouped sample values becomes simpler buta lot of information is lost, because the original sample values no longer appear explicitly.When grouping the sample values the following rules should be obeyed (Kreyszig, 1979):

All the class intervals should have the same length;

The class intervals should be chosen so that the class midpoints correspond to simple number;

If a sample valuexj coincides with the common point of two class intervals, one takes it

into the class interval that extends fromxj to the right.

1.3 Probability

Probability is an numerical measure of the chance or likelihood of occurrence of an eventrelative to other events.

Letting n to move to infinite (n ), frequencies moves to probabilities and consequentlyfj probability. Ifx 0 then histogram of normalized relative frequencies become theprobability density function (PDF) and the histogram of cumulative relative frequenciesbecome the cumulative distribution function (CDF).

A random experiment or random observation is a process that has the following properties,

(Kreyszig, 1979): it is performed according to a set of rules that determines the performance completely;

x xxmin xmax x

n1 n2 nj nm


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it can be repeated arbitrarily often;

the result of each performance depends on chance (that is, on influences which we cannotcontrol) and therefore can not be uniquely predicted.

The result of a single performance of the experiment is called the outcome of that trial. The

set of all possible outcomes of an experiment is called the sample space of the experiment andwill be denoted by S. Each outcome is called an element or point ofS.

Experience shows that most random experiments exhibit statistical regularity or stability ofrelative frequencies; that is, in several long sequences of such an experiment thecorresponding relative frequencies of an event are almost equal to probabilities. Since mostrandom experiments exhibit statistical regularity, one may assert that for any event Ein suchan experiment there is a number P(E)such that the relative frequency ofEin a great numberof performances of the experiment is approximately equal to P(E).

For this reason one postulates the existence of a number P(E)which is calledprobability of an

event E in that random experiment. Note that this number is not an absolute property ofEbutrefers to a certain sample space S, that is, to a certain random experiment.

1.4 Random variables

Roughly speaking, a random variable X (also called variate) is a function whose values arereal numbers and depend on chance (Kreyszig, 1979).

If one performs a random experiment and the event corresponding to a number a occurs, thenwe say that in this trial the random variable Xcorresponding to that experiment has assumedthe value a. The corresponding probability is denoted by P(X=a).Similarly, the probability of the event Xassumes any value in the interval a


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-4 -3 -2 -1 0 1 2 3 4x

(x)

ba

P(a a one has P(a X


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- Fx(x) >0

- As x - Fx(x)0- As x Fx(x)1- if x2 x1 Fx(x2)Fx(x1)it is an increasing function- Since for any a and b > a Fx(a) = P(X


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An asymmetric distribution does not comply with relation (1.13).

Asymmetric distributions are with positive asymmetry (skewness coefficient larger than zero)and with negative asymmetry (skewness coefficient less than zero).

A negative skewness coefficient indicates that the tail on the left side of the probabilitydensity function is longer than the right side and the bulk of the values (including the median)lies to the right of the mean.

A positive skew indicates that the tail on the right side of the probability density function islonger than the left side and the bulk of the values lie to the left of the mean. A zero valueindicates that the values are relatively evenly distributed on both sides of the mean, typicallyimplying a symmetric distribution.

Figure 1.4. Asymmetric distributions with positive asymmetry (left) and negative asymmetry(right) (www.mathwave.com)

The mode of the distribution X

is the value of the random variable that corresponds to the

peak of the distribution (the most likely value).

Distributions with positive asymmetry have the peak of the distribution shifted to the left(mode smaller than mean); distributions with negative asymmetry have the peak of thedistribution shifted to the right (mode larger than mean).

The median of the distribution, X~

, xm is the value of the random variable that have 50%chances of smaller values and, respectively 50% chances of larger values.

P(X > xm)= P(X xm) = 0.5 = Fx(xm)

For a symmetric distribution the mean, the mode and the median are coincident and theskewness coefficient is equal to zero.

The fractile xp is defined as the value of the random variable X with p non-exceedanceprobability (P(Xxp) = p).


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P(X xp) = p = Fx(xp) = Shaded arias P(X xp) = 1- p = Fx(xp) = Shaded arias

Inferior fractile Superior fractile

The loads are superior fractiles.The resistances are inferior fractiles.

If a random variableXhas mean X and variance2

x , then the corresponding variableZ = (X - X)/X has the mean 0 and the variance 1. Z is called the standardized variablecorresponding toX.

)(xFX

xxpC

umulativedistribut

ion

function

)x(F pX

1

)(xFX

xxpC

umulativedistribution

function

p

)(xfx

xxpP

robabilitydistribution

function

)(xfx

xxP

robabilitydistribution

function


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2. DISTRIBUTIONS OF PROBABILITY

2.1. Normal distribution

The continuous distribution having the probability density function, PDF

2

2

1

1

2

1)(

XXx

X

exf

(2.1)

is called the normal distribution or Gauss distribution.

A random variable having this distribution is said to be normal or normally distributed. This

distribution is very important, because many random variables of practical interest are normal

or approximately normal or can be transformed into normal random variables. Furthermore,

the normal distribution is a useful approximation of more complicated distributions.

In Equation 2.1, is the mean and is the standard deviation of the distribution. The curve off(x) is called the bell-shaped curve. It is symmetric with respect to and is biparametric (Xand x). Figure 2.1 shows f(x) for same and various values of (and various values ofcoefficient of variation V).

Normal distribution

0

0.002

0.004

0.006

0.008

0.01

0.012

100 200 300 400 500 600 x

f(x)

V=0.10V=0.20

V=0.30

Figure 2.1. PDFs of the normal distribution for various values ofV

The smaller (and V) is, the higher is the peak at x = and the steeper are the descents onboth sides. This agrees with the meaning of variance. (The inflection points of the PDFs are

at left and right ofx = ; x = + .


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From (2.1) one notices that the normal distribution has the cumulative distribution function, CDF

dvexFx v

X

X

X

2

2

11

2

1)(

(2.2)

Figure 2.2 shows F(x)for same and various values of(and various values of coefficient ofvariation V).

Normal distribution

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

100 200 300 400 500 600 x

F(x)

V=0.10

V=0.20

V=0.30

Figure 2.2. CDFs of the normal distribution for various values ofV

From (2.2) one obtains

dvaFbFbXaPb

a

v

X

e XX

2

2

11

2

1)()()(

(2.3)

In Microsoft Excel:NORMDIST(x,Xand x )normal distributionChange of variable:

The integral in (2.2) cannot be evaluated by elementary methods.

In fact, if one sets u

v

x

x

, then

xdv

du

1

or (dv = du x), and one has to integrate the

integral (2.2) from -toz =x

xx

.

)()()()()( zFzZPxX

PxXPxF zx

x

x

xx

the same type of the distribution

function. dzzfdxxfzdFxdF zxzx )()()()( and P(x X < x+dx)=P(z Z < z+dz)

From (2.2) one obtains duexFx u

X

/)(

2

2

1

2

1)( ; drops out, and the expression on the

right equals:


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duezz u

2

2

2

1)(

(2.4)

which is the distribution function of the standardized variable with mean 0 and variance 1 and

has been tabulated and wherez =x

xx

,but can be represented in terms of the integral.

X

XxxF

)( (2.5).

The density function and the distribution function of the normal distribution with mean 0 andvariance 1 are presented in Figure 2.3.

Standard normal

distribution

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

-4 -3 -2 -1 0 1 2 3 4

z

f(z)

Standard normal

distribution

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-4 -3 -2 -1 0 1 2 3 4

z

(z)

Figure 2.3. PDFand CDFof the normal distribution with mean 0 and variance 12

2

1

2

1 ze)z(f

- normal standard PDF(probability distribution function)

du)u(f)z(z

- normal standard CDF (cumulative distribution function)

In Microsoft Excel:NORMSDIST(z,z=0 and z= 1)normal standard distribution

21


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From (2.3) and (2.5) one gets:

X

X

X

Xab

aFbFbXaP

)()()( (2.6).

In particular, when a = X X and b = X X, the right-hand side equals (1) - (-1);to a = X X and b = X X there corresponds the value (2) - (-2), etc. Usingtabulated values offunction one thus finds

(a) P(X-X< X X +X)68%

(b) P(X-2X< X X +2X)95.5% (2.7).

(c) P(X-3X< X X +3X)99.7%

Hence one may expect that a large number of observed values of a normal random variableX

will be distributed as follows:

(a) About 2/3 of the values will lie between X -X and X +X

(b) About 95% of the values will lie between X -2X and X +2X

(c) About 99 % of the values will lie between X -3X and X +3X.

Practically speaking, this means that all the values will lie between X -3X and X +3X;these two numbers are called three-sigma limits.

The fractile xp that is defined as the value of the random variable Xwithp non-exceedance

probability ( P(Xxp)= p=Fx(xp)) is computed as follows:

xp= X + kpX (2.8).

The meaning ofkpbecomes clear if one refers to the reduced standard variablez = (x - )/.

Thus,x = +zand kprepresents the value of the reduced standard variable for which:

(z) =p.

Ex:x0,05 p=0,05 (non-exceedance probability)

p < 0,5kp

0,5kp>0p = 0,5kp=0we define the medianvalueThe most common values ofkpare given in Table 2.1.

Table 2.1. Values ofkpfor different non-exceedance probabilitiesp (from Tables)

p 0.01 0.02 0.05 0.95 0.98 0.99

kp -2.326 -2.054 -1.645 1.645 2.054 2.326


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2.2. Log-normal distribution

The lognormal distribution of random variable x is the normal distribution of the random

variable lnx.

The log-normal distribution (Hahn & Shapiro, 1967) is defined by its following property: if

the random variable lnX is normally distributed with mean lnX and standard deviation lnX,then the random variable X is log-normally distributed. Thus, the cumulative distribution

function CDFof random variable lnXis of normal type:

dvv

1e

1

2

1)v(lnde

1

2

1)x(lnF

xvln

2

1

Xln

xlnvln

2

1

Xln

2

Xln

Xln

2

Xln

Xln

(2.9).

Since:

x

dv)v(f)x(lnF (2.10)

the probability density function PDFresults from (2.9) and (2.10):

2

Xln

Xlnxln

2

1

Xln

ex

11

2

1)x(f

(2.11).

The lognormal distribution is asymmetric with positive asymmetry, i.e. the peak of the

distribution is shifted to the left. The skewness coefficient for lognormal distribution is:

3

1 3 XX VV (2.12)

where VX is the coefficient of variation of random variable X. Higher the variability, higher

the shift of the lognormal distribution.

The mean and the standard deviation of the random variable lnXare related to the mean and

the standard deviation of the random variableXas follows:

21 X

XXln

V

ln

(2.13)

)Vln( XXln21 (2.14).

In the case of the lognormal distribution the following relation between the mean and the

median holds true:

50.Xln xln (2.15).

Combining (2.13) and (2.15) it follows that the median of the lognormal distributions is

linked to the mean value by the following relation:

250

1 X

X.

V

x

(2.16)


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IfVXis small enough (VX0.1), then:

XXln ln (2.17)

XX Vln (2.18)

The PDF and the CDF of the random variable X are presented in Figure 2.4 for differentcoefficients of variation.

Log-normal distribution

0

0.002

0.004

0.006

0.008

0.01

0.012

100 200 300 400 500 600 x

f(x)

V=0.10

V=0.20

V=0.30

Log-normal distribution

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

100 200 300 400 500 600x

F(x)

V=0.10

V=0.20

V=0.30

Figure 2.4. Probability density function,f(x)and cumulative distribution function, F(x)of the log-normal distribution for various values ofV

The mode median mean of the distribution.


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Change of variable:

If one uses the reduced variable uv

v

v

ln

lnln

, thenvdv

du

v

11

ln

or (dv = du v lnv),

and one has to integrate from -toz =v

vvln

lnln . From (2.9) one obtains:

dueduvv

ezz u

v

v u

v

vv

2ln

/)(ln

2

ln

2lnln

2

2

111

2

1)(

(2.19)

Standard normal cumulative distribution function (CDF)

The fractile xp that is defined as the value of the random variable Xwith p non-exceedance

probability (P(Xxp)= p) is computed as follows, given lnXnormally distributed:

ln(xp) = lnX+ kplnX (2.20)From (2.20) one gets:

XpX k

p exlnln (2.21)

where kprepresents the value of the reduced standard variable for which (z) =p.

Xlnexx ,m 50 ; Xln.xln 50 relation between the median and the mean

Normal distribution for X :

X

X

X

Xab

aFbFbXaP

)()()(

LogNormal distribution for X:

xln

xln

xln

xln alnbln)a(F)b(F)bXa(P

2.3. Extreme value distributions

The field of extreme value theory was pioneered by Leonard Tippett (19021985). Emil

Gumbel codified the theoryof extreme values in his book Statistics of extremes published

in 1958 at Columbia University Press.

Extreme value distributions are the limiting distributions for the minimum or the maximum of

a very large collection of random observations from the same arbitrary distribution. Theextreme value distributions are of interest especially when one deals with natural hazards like

snow, wind, temperature, floods, etc. In all the previously mentioned cases one is not

interested in the distribution of all values but in the distribution of extreme values, which

might be the minimum or the maximum values. In Figure 2.5 it is represented the distribution

of all values of the random variableXas well as the distribution of minima and maxima ofX.

Year 1: y11, y21, y31, ym1 - x1Year 1: y12, y22, y32, ym2 - x2..

Year n: y1n

, y2n

, y3n

, ymn

- xnXi= max (y1i, y2i, y3i, ymi)maximum annual value


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0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

x

f(x)

all values

distribution

minima

distribution

maxima

distribution

Figure 2.5. Distribution of all values, of minima and of maxima of random variable X

2.3.1. Gumbel distribution for maxima in 1 year

The Gumbel distribution for maxima is defined by its cumulative distribution function, CDF:

)ux(ee)x(F

(2.22)

where uand are the parameters:

u = x

; = 0,5772Euler constant (Euler-Mascheroni);

n

kn

nlnk

lim1

1

x

1

6

The final form is: u = x0.45xmode of the distribution (Figure 2.8) and

= 1.282 /xdispersion coefficient (shape factor).

The skewness coefficient of Gumbel distribution is positive constant ( 139.11 ), i.e. thedistribution is shifted to the left. In Figure 2.6 it is represented the CDF of Gumbel

distribution for maxima for the random variable X with the same mean x and differentcoefficients of variation Vx.The probability distribution function, PDFis obtained straightforward from (2.22):

)ux(e)ux(

eedx

)x(dF)x(f

(2.23)

The PDFof Gumbel distribution for maxima for the random variable Xwith the same mean

xand different coefficients of variation Vxis represented in Figure 2.7.


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Gumbel distribution

for maxima in 1 year

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

100 200 300 400 500 600 700

x

F(x)

V=0.10

V=0.20V=0.30

Figure 2.6. CDFof Gumbel distribution for maxima for the random variableX

with the same mean xand different coefficients of variation Vx

Gumbel distribution

for maxima in 1 year

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

100 200 300 400 500 600 700

x

f(x)

V=0.10

V=0.20

V=0.30

Figure 2.7. PDFof Gumbel distribution for maxima for the random variableX

with the same mean xand different coefficients of variation Vx

One can notice in Figure 2.7 that higher the variability of the random variable, higher the shiftto the left of the PDF.


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0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.0045

0.005

100 200 300 400 500 600 700

x

(x)

0.45 x

u x

Figure 2.8. Parameter uin Gumbel distribution for maxima

Fractilexp:

GivenXfollows Gumbel distribution for maxima, the fractilexpthat is defined as the value of

the random variable X with p non-exceedance probability (P(X xp) = p) is computed asfollows:

)upx(e

pp ep)xX(P)x(F

(2.24).

From Equation 2.24 it follows: )plnln()ux(e)pln( p)ux( p

x

G

px

x

xxp k)plnln(282.1

45.0)plnln(1

ux

(2.25)

where:

)plnln(78.045.0kG

p (2.26).

The values ofkpGfor different non-exceedance probabilities are given in Table 2.2.

Table 2.2. Values ofkpGfor different non-exceedance probabilitiesp

p 0.50 0.90 0.95 0.98

kpG -0.164 1.305 1.866 2.593

xm=x - 0,164x(median value)

xm

0,164 x


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2.3.2. Gumbel distribution for maxima in N years

All the previous developments are valid for the distribution of maximum yearly values. If one

considers the probability distribution in N (N>1) years, the following relation holds true (if

one considers that the occurrences of maxima are independent events):

F(x)1 year= P(Xx)in 1 year

F(x)N years= P(Xx)in N years= P(Xx)year1 P(Xx)year2 P(Xx)year N

F(x)N years= P(Xx)in N years = [P(Xx)in 1 year]N= [F(x)1 year]N (2.27)

where:

F(x)N yearsCDFof random variableXinNyears

F(x)1 yearCDFof random variableXin 1year.

The Gumbel distribution for maxima has a very important property the reproducibility of

Gumbel distribution - i.e., if the annual maxima (in 1 year) follow a Gumbel distribution for

maxima then the maxima inNyears will also follow a Gumbel distribution for maxima:

)ux()ux( Ne

NeN

N ee)x(F)x(F1111

1

)Nux(N))

Nlnu(x(

Nln)ux(eee eee

1

1111

(2.28)

where:

u1mode of the distribution in 1 year

1dispersion coefficient in 1 year

uN= u1+ lnN /1mode of the distribution inNyears = 1dispersion coefficient inNyearsThe PDFof Gumbel distribution for maxima inNyears is translated to the right with the amount

lnN /1 with respect to the PDFof Gumbel distribution for maxima in 1 year, Figure 2.9.

Gumbel distribution

for maxima

0

0.0021

0.0042

0.0063

100 200 300 400 500 600 700 800 900 1000

x

f(x)

N yr.

1 yr.

u 1 uN

lnN / 1

m 1 mN

lnN / 1

Figure 2.9. PDFof Gumbel distribution for maxima in 1 year and in Nyears


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11

,x

x,xV

;

N,x

xN,xV

; Vx,N < Vx,1(COV is decreasing with increasing the years)

Also, the CDF of Gumbel distribution for maxima in N years is translated to the right with the

amount lnN /1 with respect to the CDFof Gumbel distribution for maxima in 1 year, Figure 2.10.

Gumbel distribution

for maxima

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

F(x)

N yr.

1 yr.

Figure 2.10. CDFof Gumbel distribution for maxima in 1 year and in Nyears

Important notice: the superior fractile xp (p >> 0.5) calculated with Gumbel distribution formaxima in 1 year becomes a frequent value (sometimes even an inferior fractile ifNis large,

N50) if Gumbel distribution for maxima inNyears is employed, Figure 2.11.

Gumbel distribution

for maxima

0

0.0021

0.0042

0.0063

x

f(x)

N yr.

1 yr.

probability of

exceedance ofx p

inNyears

probability of

exceedance ofx p

in 1 year

x p

Figure 2.11. Superior fractilexpin 1 year and its significance inNyear


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2.4. Mean recurrence interval

The loads due to natural hazards such as earthquakes, winds, waves, floods were recognized

as having randomness in time as well as in space. The randomness in time was considered in

terms of the return period or recurrence interval. The recurrence interval also known as a

return period is defined as the average (or expected) time between two successive statisticallyindependent events and it is an estimate of the likelihood of events like an earthquake, flood

or river discharge flow of a certain intensity or size. It is a statistical measurement denoting

the average recurrence interval over an extended period of time. The actual time Tbetween

events is a random variable.

The mean recurrence interval,MRIof a value larger thanxof the random variable Xmay be

defined as follows:

pxFxXPxXP)xX(MRI

xyearyear

1

1

1

1

1

11

11

(2.29)

where:

pis the annual probability of the event (Xx) and FX(x) is the cumulative distribution function ofX.

Thus the mean recurrence interval of a value x is equal to the reciprocal of the annual

probability of exceedance of the valuex. The mean recurrence interval or return period has an

inverse relationship with the probability that the event will be exceeded in any one year.

For example, a 10-year flood has a 0.1 or 10% chance of being exceeded in any one year and

a 50-year flood has a 0.02 (2%) chance of being exceeded in any one year. It is commonly

assumed that a 10-year earthquake will occur, on average, once every 10 years and that a 100-

year earthquake is so large that we expect it only to occur every 100 years. While this may be

statistically true over thousands of years, it is incorrect to think of the return period in this

way. The term return period is actually misleading. It does not necessarily mean that the

design earthquake of a 10 year return period will return every 10 years. It could, in fact, never

occur, or occur twice. This is why the term return period is gradually replaced by the term

recurrence interval. Researchers proposed to use the term return period in relation with the

effects and to use the term recurrence interval in relation with the causes.

The mean recurrence interval is often related with the exceedance probability in Nyears. The

relation amongMRI,Nand the exceedance probability inNyears, Pexc,N

is:

Pexceedance, N (>x) = 1Pnon-exceedance, N (x) = 1 - [Pnon-exceedance, 1 (x)]N=1-pN

RPreturn period

Values of a randomvariable (natural hazard)

x

years

RP it is not the same(random variable)


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Pexceedance, N (>x) = 1 - TNN

exT

1

)(

11 (2.30)

pxXPxXMRI

year

p

1

1

1

1)(

1

; )( pxXMRI = )( pxXT ;

example: x0.98similar 50Tx ; yearsMRI 5098,01

1

Usually the number of years, N is considered equal to the lifetime of ordinary buildings, i.e.

50 years. Table 2.3 shows the results of relation (2.30) for some particular cases considering

N=50 years.

Table 2.3 Correspondence amongstMRI, Pexc,1 yearand Pexc,50 years

Mean recurrence interval,

yearsMRI, )( xT

Probability of exceedance

in 1 year, Pexc,1 year

Probability of exceedance

in 50 years, Pexc,50 years10 0.10 0.99

30 0.03 0.81

50 0.02 0.63

100 0.01 0.39

225 0.004 0.20

475 0.002 0.10

975 0.001 0.05

2475 0.0004 0.02

The modern earthquake resistant design codes consider the definition of the seismic hazard

level based on the probability of exceedance in 50 years. The seismic hazard due to ground

shaking is defined as horizontal peak ground acceleration, elastic acceleration response

spectra or acceleration time-histories. The level of seismic hazard is expressed by the mean

recurrence interval (mean return period) of the design horizontal peak ground acceleration or,

alternatively by the probability of exceedance of the design horizontal peak ground

acceleration in 50 years. Four levels of seismic hazard are considered in FEMA 356

Prestandard and Commentary for the Seismic Rehabilitation of Buildings, as given in Table

2.4. The correspondence between the mean recurrence interval and the probability of

exceedance in 50 years, based on Poisson assumption, is also given in Table 2.4.

Table 2.4. Correspondence between mean recurrence interval and probability of exceedance

in 50 years of design horizontal peak ground acceleration as in FEMA 356

Seismic HazardLevel

Mean recurrence interval(years)

Probability ofexceedance

SHL1SHL2SHL3SHL4

72225475

2475

50% in 50 years20 % in 50 years10 % in 50 years2 % in 50 years


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3. FUNCTION OF RANDOM VARIABLES

3.1 Second order moment models

Let us consider a simply supported beam, Figure 3.1:

l

q

Figure 3.1. Simple supported beam

Considering that:

yq , are described probabilistically (normal, lognormal, Gumbel for maxima distributions)

and l, Ware described deterministically, the design condition (ultimate limit state condition)

is:

capMM max ; Wql

y

8

2

and considering

Ssectional effect of load, and

Rsectional resistance

it follows that:

8

2ql

S ; WR y .

The following question rises:

If q and y are described probabilistically, how can one describes SandRprobabilistically?

If X1, X2, Xi,.. Xn are random variables defined by )(1

xfx , )(2 xfx )(xf nx ;

1xm ,

2xm

nxm ;

1x , 2x nx

If Y (X1, X2, Xi,.. Xn) is the function of random variables how to find fy, ym , y

To answer the question, two cases are considered in the following:

1. The relation between qand S( y andR) is linear

2. The relation is non linear.

Case 1: Linear relation between random variables, X, Y

bXaYX (aand bconstants,Xrandom variable)


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For the random variable Xone knows: the probability density function, PDF, the cumulative

distribution function, CDF, the mean, m and the standard deviation, . The unknowns are thePDF, CDF, m and for the random variableY.

If one applies the equal probability formula, Figure 3.2 (a linear transformation keeps the

distribution):

)()()()( dyyYyPdxxXxPdyyfdxxf YX (3.1)

a

byx

;

a

byf

axf

a

dx

dyxfyf XXXY

1)(

11)()( (3.2)

Figure 3.2. Linear relation between random variablesXand Y

Distribution of YDistribution of X

Developing further the linear relation it follows that:

bmadxxfbdxxxfadxxfbxadyyyfm XXXXYY )()()()()(

dxfbmabxadyyfmy XXYYY222 )()()(

2222XXX adx)x(f)mx(a

bmam

a

XY

XY

(3.3)

bma

a

mV

X

X

Y

YY

y=ax+b

x dxx

y

dyy

dxxX )(

dyyY )(

)(xX

?)( yYx

y


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Case 2: Non-linear relation between random variables, X, Y

Let the random variable ),,,,( 21 ni XXXXYY , the relation being non-linear, how to

findfy, ym , y

Observations:

1. Let the random variable nXXXXY 321 . If the random variablesXi

are normally distributed then Yis also normally distributed.

2. Let the random variable XXXXY 321 . If the random variablesXiare

log-normally distributed then Yis also log-normally distributed.

Let the random variablesXiwith known mean and standard deviations:

nim

X

i

i

X

X

i ,1;

....)(!2

)()(

!1

)()()( 2

ax

afax

afafxf Taylor series

The first order approximation

FORMFirst order reliability method

If one develops the function Y in Taylor series around the mean m

=(1x

m ,2x

m ix

m ..nx

m ), and keeps only the first order term of the series, then the mean

and the standard deviation of the new random variable Y, YYm , are approximated by:

),,,,( 21 ni XXXXYY Y( 1xm , 2xm ixm . nxm ) + ...)(1

iXi

n

i mi

mXX

Ylinear

relation

Ym Y ( 1xm , 2xm ixm .. nxm ) - First order approximation of mean (3.4)

2

y 2

2

1iX

m

n

i iX

Y

- First order approximation of the variance

n

i iX

minX

mnX

mX

mY x

y

x

y

x

y

x

y

1

22

22

2

2

2

2

2

1

2

1

2

(3.5)

Relations 3.4 and 3.5 are the basis for the so-called First Order Second Moment Models,FOSMM.Few examples ofFOSMMare provided in the following:

21 XXY

Ym Y ( 1xm , 2xm ) = 21 XX mm

;11 2

2

2

1

2

2

22

1

22

2

2

2

2

1

2

1

2XXXXX

m

X

m

Y

x

y

x

y

2

2

2

1 XXY


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bXaY bmam XY

222XY a ; XY a

21 XXY Ym Y ( 1xm , 2xm ) = 21 XX mm

22

2

1

2

1

2

2

2

2

2

2

2

1

2

1

2

2

2

2

2

1

2

1

2

XXXXXmXmX

m

X

m

Y mmxxx

y

x

y

2

2

2

12

2

2

22

1

2

1

21

2

2

2

1

2

1

2

2

XX

X

X

X

X

XX

XXXX

Y

YY VV

mmmm

mm

mV


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4. STRUCTURAL RELIABILITY ANALYSIS

4.1. The basic reliability problem

The basic structural reliability problem considers only one load effect S resisted by oneresistance R. Each is described by a known probability density function, fS( ) and fR( )

respectively. It is important to notice thatRand Sare expressed in the same units.

For convenience, but without loss of generality, only the safety of a structural element will beconsidered here and as usual, that structural element will be considered to have failed if itsresistance R is less than the load effect S acting on it. The probability of failure Pf of thestructural element can be stated in any of the following ways, (Melchers, 1999):

Pf= P(RS) (4.1a)

=P(R-S0) (4.1b)

=P(R/S1) (4.1c)

=P(lnR-lnS0) (4.1d)or, in general

=P(G(R ,S)0) (4.1e)

where G( ) is termed the limit state functionand the probability of failure is identical with the

probability of limit state violation.

Re= P(R>S) - probability of reliability

Pf+ Re=1

(R>Ssafe; R=Slimit state; R


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In Figure 4.1, the Equations (4.1) are represented by the hatched failure domain D, so that theprobability of failure becomes:

dsdrsrfSRPPD

RSf ,0 (4.2).

WhenRand Sare independent,fRS(r,s)=fR(r)fS(s) and relation (3.2) becomes:

dxxfxFdsdrsfrfSRPP SRrs

SRf 0 (4.3)

Relation (4.3) is also known as a convolution integral with meaning easily explained by

reference to Figure 4.2. FR(x) is the probability that Rx or the probability that the actualresistanceRof the member is less than some valuex. Let this represent failure. The termfs(x)represents the probability that the load effect Sacting in the member has a value between x

andx+xasx0. By considering all possible values ofx, i.e. by taking the integral over allx, the total probability of failure is obtained. This is also seen in Figure 4.3 where the densityfunctionsfR(r) andfS(s) have been drawn along the same axis.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

- - - -x

FR (x), fS(x)

R

S

R=x

P(R


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IfSis distanced fromRwe deal with Pfsmall.

IfSis near toRwe deal with Pflarge.

An alternative to expression (4.3) is:

dxxfxFP RSf )()(1 (4.4)

which is simply the sumof the failure probabilities over all cases of resistance for which the

load effect exceeds the resistance.

4.2. Special case: normal random variables

For a few distributions ofR and S it is possible to integrate the convolution integral (4.3)analytically.

One notable example is when both are normal random variables with means R and Sandvariances R2and S2respectively.

The safety marginZ=R-S(Cornell approach) follow an normal distribution with a mean andvariance given by :

Z= R - S (4.5a)

Z2= R2+ S2 22 SRZ (4.5b)

Equation (4.1b) then becomes

1

0000

Z

ZZf )z(FZPSRPP (4.6)

where ( ) is the normal standard cumulative distribution function (for the standard normalvariate with zero mean and unit variance).

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

-60 -50 -40 -30 -20 -10 0 10 20z

Z(z) z

z0

P

Z0

Failure Safety

Figure 4.4. Distribution of safety marginZ = RS

0r

s

Failure domain(z0)

z = r - s = 0Limit statefunction


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The random variable Z = R - S is shown in Figure 4.4, in which the failure region Z0 isshown shaded. Using (4.5) and (4.6) it follows that (Cornell, 1969):

1

0

22

SR

SRfP (4.7)

where =z

z

is defined as reliability (safety) index.

If either of the standard deviations Rand Sor both are increased, the term in square bracketsin (4.7) will become smaller and hence Pfwill increase. Similarly, if the difference between the

mean of the load effect and the mean of the resistance is reduced, Pf increases. These

observations may be deduced also from Figure 4.3, taking the amount of overlap offR( ) andfS()

as a rough indicator ofPf.

Pf 10-3 3.09

10-4 3.72

10-5 4.27

10-6 4.75

4.3. Special case: log-normal random variables

The log-normal model for structural reliability analysis was proposed by (Rosenbluth &Esteva, 1972). Both random variables R and S have lognormal distribution with the following

parameters: means lnRand lnSand variances lnR2

and lnS2

respectively.The safety marginZ=

S

Rln then has a mean and a standard deviation given by:

S

R

S

RZ

lnln

(4.8a)

22

lnSR

S

R

S

RZ VVV (4.8b)

R

R

RR

V

ln

1ln

2ln

; S

S

SS

V

ln

1ln

2ln

RRR VV )1ln( 2ln ; SSS VV )1ln( 2lnRelation (4.1d) then becomes

Z

Zf ZP

S

RPP

0

00ln (4.9)

where ( ) is the standard normal distribution function (zero mean and unit variance). The

random variable Z=S

Rln is shown in Figure 4.5, in which the failure regionZ0 is shown

shaded.

Using (4.8) and (4.9) it follows that


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1

0

22SR

S

R

f

VV

ln

P (4.10)

where =z

z

is defined as reliability (safety) index,

22

ln

SR

S

R

VV

. (4.11)

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

-60 -50 -40 -30 -20 -10 0 10 20z

Z(z) z

z0

P

Z0

Failure Safety

Figure 4.5. Distribution of safety marginZ=

S

Rln

4.4. Calibration of partial safety coefficients

In USA:LRDFLoad and resistance factor design

22

ln

SR

S

R

VV

Lindt proposed the following liniarization: RSRS VVVV 22 with 75,07,0 for

33

1

S

R

V

V. Given Lindts linearization it follows that:


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SRS

R

VV

ln

(4.12).

The calibration of partial safety coefficients used in semi-probabilistic design codes isaccomplished using the log-normal model using FOSM.From Equation 4.12 one has:

SRSR VV

S

RVV

S

R

SR

S

R

eeeVV

ln

SR V

S

V

R ee (4.13)

where RVe and SVe are called safety coefficients, SCwith respect to the mean.

Rqand Spare characteristics values (fractiles) of the random variablesRrespectively S.

RqRqq Vk

R

VkRRkRq eeeeR

lnlnln

SpSpp Vk

S

VkSSkSp eeeeS

lnlnln

RVk

SVk

Vk

VkVRV

Vk

Vk

S

R

p

q

q

p

Sp

RqS

Sp

Rq

e

e

e

ee

e

e

S

R

)(

)()(

p

SVk

p

q

RVk

qpq eSeR

)()(

RVk

qqe

)( - Partial safety coefficient for sectional resistanceSVk

ppe

)(

- Partial safety coefficient for the effects of the load

But one needs the SCwith respect to the characteristic values of the loads and resistances, theso-called partial safety coefficients, PSC.kq= -1,645 (p=0.05) k p= 1,645 (p=0.95)kq= -2,054 (p=0.02) k q= 2,054 (p=0.98)

To this aim, one defines the limit state function used in the design process:

designdesignR

R

S

S 05.005.098.098.0 (4.14)

where 98.0 and 05.0 are called partial safety coefficients, PSC.

Assuming thatSandRare log-normally distributed, one has:

RR V

R

VRRR eeeeR 645.1645.1lnln645.1ln05.0

(4.15)

SS V

S

VSSS eeeeS 054.2054.2lnln054.2ln98.0

(4.16)

S

R

S

R

V

VSVRV

V

V

S

R

e

eee

e

e

S

R

054.2

645.1

054.2

645.1

98.0

05.0

(4.17)


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98.0

)054.2(

98.0

05.0

)645.1(

05.0

SVRV eSeR (4.18)

RVe)645.1(

05.0

- Partial safety coefficient for sectional resistance (4.19)

SVe )054.2(98.0 - Partial safety coefficient for the effects of the load (4.20)

The partial safety coefficients 05.0 and 98.0 as defined by Equations 4.19 and 4.20 depend onthe reliability index and the coefficients of variation for resistances and loads, respectively.If the reliability index is increased, the partial safety coefficient for loads 98.0 increaseswhile the partial safety coefficient for resistance 05.0 decreases. The theory of partial safetycoefficients based on lognormal model is incorporated in the Romanian Code CR0-2013named Cod de proiectare. Bazele proiectarii structurilor in constructii (Design Code. Basisof Structural Design).

1.0

1.2

1.4

1.6

1.8

2.0

0 0.2 0.4 0.6

g 9 8

VS0.6

0.7

0.8

0.9

0.05 0.1 0.15 0.2

0.05

VR

Figure 4.6. Variation of partial safety coefficients with the coefficient of variation of loadeffect (left) and of resistance (right)

Eurocode 1 (EN1-1990)Basis of Structural design Rd Ed(design condition)

= 4,7ultimate Limit states (ULS)

= 2,9Serviceability Limit states (SLS)

Actions (F) : Permanent actions (G), Variable actions (Q), Accidental action (A)Fkcharacteristic value (upper fractile)Fddesign value

f - partial safety coefficient applied to the actionFd= f FkEd= Sd E(Fd) - design value of section effort of the loadE(Fd)sectional effect of the design action (load)

Sd - safety coefficient considering the errors involved in the model for calculation of thesectional effort of the load


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Ed= E( Sd f Fk) = E( F Fk)

Xkcharacteristic value of the strength of the material (inferior fractile)Xddesign value of the strength of the material

m - partial safety coefficient applied to the strength of the material

Xd= km

X1

Rddesign value of the sectional resistance

Rd= )X(R)X(R)X(R kM

kmR

dR dd

1111

Other representative values of variable actions used in the design, Figura B.4.1are:

the combination value, represented as a product of0Qk used for the verification ofultimate limit states and irreversible serviceability limit states;

the frequent value, represented as a product 1Qk, used for the verification of ultimatelimit states involving accidental actions and for verifications of reversibleserviceability limit states; this value is closed to the central value of the statisticaldistribution;

the quasi-permanent value, represented as a product 2Qk, (2 1) used for theverification of ultimate limit states involving accidental actions and for the verificationof reversible serviceability limit states. Quasi-permanent values are also used for thecalculation of long-term effects.

0

5

10

15

20

25

Timp

Valoare

instantaneeQ

Valoare caracteristica, Qk

Valoare de combinatie, 0Qk

Valoare frecventa, 1Qk

Valoare cvasipermanenta, 2Qk

1234

fQ

Figure 4.7 Values of variable actions

Ed = i,ki,m

ii,Q,k,Qpj,k

n

jj,G QQPG 0

211

1

- fundamental combination

Ed = i,ki,m

i

Edj,k

n

j

QAPG 211

- seismic combination


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5. SEISMIC HAZARD ANALYSIS

The final product of the seismic hazard analysis is the hazard curve for a given site.

PSHA - Bucharest

1.E-04

1.E-03

1.E-02

1.E-01

100 150 200 250 300 350 400

PGA, cm/s2

Annua

lexceedancerate,

(PGA)

Hazard curve for Bucharest from Vrancea seismic source

y

yMRI 1)( ; ymean annual rate of ground exceedance of mean parameter (PGA-peak

ground acceleration, SA-spectral acceleration, peak ground displacement, etc)The two methods of seismic hazard analysis are deterministic and probabilistic ones.

5.1. Deterministic seismic hazard analysis (DSHA)

The deterministic seismic hazard analysis involves the development of a particular seismicscenario, i.e. the postulated occurrence of an earthquake of a specified size at a specificlocation (longitude, latitude, magnitude and depth are deterministic). The DSHA isdeveloped in four steps (Reiter, 1990):

1. Identification and characterization of all earthquake sources geometry and positionof the sources and the maximum magnitude for all sources;

M- Source (focus)

Rhypocentraldistance

hdepth

Epicenter Site- epicentraldistance


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2. Selection of source-to-site distance parameters (either epicentral or hypocentraldistance) for each source zone;

3. Selection ofcontrollingearthquake (expected to produce the strongest shaking at thesite); use predictive (attenuation) relations for computing ground motion produced atthe site by earthquakes of magnitudes given in step 1 occurring at each source zone;

4. Define the seismic hazard at the site in terms of peak ground acceleration PGA,spectral acceleration SA, peak ground velocity PGV, etc.

All the steps are summarized in Figure 5.1.

NY

Y

Y

Y

Y

3

2

1

Figure 5.1. Steps inDSHA

5.2. Probabilistic seismic hazard analysis (PSHA)

The PSHA(Cornell, 1968, Algermissen et. al., 1982) is developed in four steps (Reiter, 1990):1. Identification and characterization of earthquake sources. Besides the information

required in step 1 ofDSHA, it is necessary to obtain the probability distribution ofpotential rupture location within the source and the probability distribution of sourceto-site distance (longitude, latitude, magnitude and depthare random variable);

Site

Source 1M1

Source 3M3

Source 2M2

R1

R3

R2

Step 1 Step 2

Distance

GroundMotionParameter,Y

R2 R3 R1

M

M

M

Step 3Step 4


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2. Definition of seismicity, i.e. the temporal distribution of earthquake recurrence(average rate at which an earthquake of some size will be exceeded);

3. Use predictive (attenuation) relations for computing ground motion produced at thesite by earthquakes of any possible size occurring at any possible point in each sourcezone; uncertainty in attenuation relations is considered in PSHA;

4. Uncertainties in earthquake location, size and ground motion prediction are combinedand the outcome is the probability that ground motion parameter will be exceededduring a particular time period.

All the steps are summarized in Figure 5.2.

Figure 5.2. Steps in PSHA

5.3. Earthquake source characterization

The seismic sources can be modeled as:

point sourcesif the source is a short shallow fault

area sourcesif the source is a long and/or deep fault volumetric sources.

Site

Source 1M1

Source 3M3

Source 2M2

Step 1 Step 2

Step 3 Step 4

Magnitude

Averagerate,log

2

3

1

Distance, R

GroundMotion

Parameter,Y

R

f

R

f

R

f


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The spatial uncertainty of earthquake location is taken into account in PSHA. The earthquakesare usually assumed to be uniformly distributed within a particular source zone. The uniformdistribution in the source zone does not often translate into a uniform distribution of source-to-site distance.Another important source of uncertainties is given by the size of the earthquake and by the

temporal occurrence of earthquakes. The recurrence law gives the distribution of earthquakesizes in a given period of time.Gutenberg & Richter (1944) organized the seismic data in California according to the numberof earthquakes that exceeded different magnitudes during a time period. The key parameter in

Gutenberg & Richters work was the mean annual rate of exceedance, M of an earthquake ofmagnitude Mwhich is equal to the number of exceedances of magnitude Mdivided by thelength of the period of time. The Gutenberg & Richter law is (Figure 5.3):

lg M= a - b M (5.1)

where M-mean annual rate of exceedance of an earthquake of magnitudeM,M- magnitude,aand bnumerical coefficients depending on the data set.

If NMis the number of earthquakes with magnitude higher than M in T years M=T

NM

The physical meaning ofaand bcoefficients can be explained as follows:

0 =10a mean annual number of earthquakes of magnitude greater than or equal to 0 (a

gives the global seismicity)b describes the relative likelihood of large to small earthquakes. If b increases the

number of larger magnitude earthquakes decreases compared to those of smaller earthquakes(bis the slope of the recurrence plot).

Figure 5.3. The Gutenberg-Richter law

The aand bcoefficients are obtained through regression on a database of seismicity form thesource zone of interest. Record of seismicity contains dependent events (foreshocks,aftershocks) that must be removed form the seismicity database because PSHAis intended to

evaluate the hazard form discrete, independent releases of seismic energy.

0 M

lg M10

a

b

bsmall

blarge

1


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lg M=10ln

ln M = a - b M ; M= Mba10 (5.2)

ln M= a ln10 - b ln10 M = - M (5.3)

M=M

e

(5.4)

where = a ln10 = 2.303 a and = b ln10 = 2.303 b.

The original Gutenberg & Richter law (5.1) is unbounded in magnitude terms. This leads tounreliable results especially at the higher end of the magnitude scale. In order to avoid thisinconsistency, the bounded recurrence law is used.The bounded law is obtained and defined hereinafter.

The form (5.4) of Gutenberg & Richter law shows that the magnitudes follow an exponentialdistribution. If the earthquakes smaller than a lower threshold magnitude Mminare eliminated,one gets (McGuire and Arabasz, 1990):

FM(M) = P[Mag. MMMmin] = 1 - P[Mag. > MMMmin]=

=

minM

M1

= 1 -

minM

M

e

e

= 1 -

)MM( mine (5.5)

fM(M) =dM

)M(dFM = min)MM(e . (5.6)

Mminis the mean annual rate of earthquakes of magnitude Mlarger or equal thanMmin.If both a lower threshold magnitude Mmin and a higher threshold magnitude Mmax are takeninto account, the probabilistic distribution of magnitudes can be obtained as follows (McGuireand Arabasz, 1990); (Bounded Guttenberg-Richter law).The cumulative distribution function must have the unity value forM = Mmax. This yields:

)(MFM = P[Mag.MMminMMmax] =)M(F

)M(F

maxM

M =)MM(

)MM(

minmax

min

e1

e1

(5.7)

)(MfM =

dM

MdFM )( =)MM(

)MM(

minmax

min

e1

e

. (5.8)

The mean annual rate of exceedance of an earthquake of magnitudeMis:

)](1[min

MFMMM = )MM(

)MM()MM(

Mminmax

minmaxmin

mine1

ee

(5.9)

whereminM

= minMe is the mean annual rate of earthquakes of magnitude Mlarger or

equal thanMmin.Finally one gets (McGuire and Arabasz, 1990):


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M = minM

e

)MM(

)MM()MM(

minmax

minmaxmin

e1

ee

=

= minMe )MM(

)MM()MM(

minmax

maxmin

e1

]e1[e

= Me

)MM(

)MM(

minmax

max

e1

e1

(5.10)

M =M

e )MM(

)MM(

minmax

max

e1

e1

5.4. Predictive relationships (attenuation relations)

The predictive relationships usually take the form Y=f(M, R, Pi), where Yis a ground motionparameter, M is the magnitude of the earthquake, R is the source-to-site distance and Piareother parameters taking into account the earthquake source, wave propagation path and siteconditions. The predictive relationships are used to determine the value of a ground motion

parameter for a site given the occurrence of an earthquake of certain magnitude at a givendistance. The coefficients of the predictive relationships are obtained through least-squareregression on a particular set of strong motion parameter data for a given seismic region. Thisis the reason for not extrapolating the results of the regression analysis to another seismicregion. The uncertainty in evaluation of the ground motion parameters is incorporated inpredictive relationships through the standard deviation of the logarithm of the predictedparameter. Finally, one can compute the probability that ground motion parameter Yexceeds acertain value,y*for an earthquake of magnitude, mat a given distance r(Figure 5.4):

*1,|*1,|* yFrmyYPrmyYP Y (5.11)where Fis the CDF of ground motion parameter, usually assumed lognormal.

Y

R

y*

P(Y>y*|m,r)

r

Y(y|m,r)

Figure 5.4. Incorporation of uncertainties in the predictive relationships

5.5. Temporal uncertainty

The distribution of earthquake occurrence with respect to time is considered to have a randomcharacter. The temporal occurrence of earthquakes is considered to follow, in most cases, aPoisson model, the values of the random variable of interest describing the number ofoccurrences of a particular event during a given time interval.

The properties of Poisson process are:


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1. The number of occurrences in one time interval is independent of the number ofoccurrences in any other time interval.

2. The probability of occurrence during a very short time interval is proportional to thelength of the time interval.

3. The probability of more than one occurrence during a very short time interval is

negligible.IfN is the number of occurrences of a particular event during a given time interval, theprobability of having noccurrences in that time interval is:

!n

enNP

n (5.12)

where is the average number of occurrences of the event in that time interval.

5.6. Probability computations

The results of the PSHAare given as seismic hazard curvesquantifying the annual probability

of exceedance of different values of selected ground motion parameter.The probability of exceedance a particular value,y*of a ground motion parameter (GMP) Yiscalculated for one possible earthquake at one possible source location and then multiplied bythe probability that that particular magnitude earthquake would occur at that particularlocation. The process is then repeated for all possible magnitudes and locations with theprobability of each summed:

dxxfXyYPXPXyYPyYP X|*|** (5.13)whereXis a vector of random variables that influence Y(usually magnitude,Mand source-to-site distance, R). Assuming M and R independent, for a given earthquake recurrence, theprobability of exceeding a particular value, y*, is calculated using the total probabilitytheorem (Cornell, 1968, Kramer, 1996):

dmdrrfmfrmyYPyYP RM )()(),|*(*)( (5.14)where:- P(Y>y*|m,r) probability of exceedance ofy* given the occurrence of an earthquake ofmagnitude mat source to site distance r.-fM(m)probability density function for magnitude;

- fR(r)probability density function for source to site distance.

The attenuation relation for subcrustal earthquakes (Mollas & Yamazaki, 1995):

ln PGA = c0+ c1Mw+ c2lnR +c3R +c4h +

where: PGAis peak ground acceleration at the site,

Mw- moment magnitude,R - hypocentral distance to the site,

M- Source (focus)

Rhypocentraldistance

hdepth

Epicenter Site- epicentraldistance


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h - focal depth,c0, c1, c2, c3, c4 - data dependent coefficients

- random variable with zero mean and standard deviation = ln PGA.

dRdM)R(f)M(f)R,M|*PGAPGA(P*)PGAPGA(P wRwMw w

5.7. Probabilistic seismic hazard assessment for Bucharest from Vrancea seismic source

The Vrancea region, located when the Carpathians Mountains Arch bents, is the source ofsubcrustal (60-170km) seismic activity, which affects more than 2/3 of the territory ofRomania and an important part of the territories of Republic of Moldova, Bulgaria andUkraine. According to the 20thcentury seismicity, the epicentral Vrancea area is confined to arectangle of 40x80km2having the long axis oriented N45E and being centered at about 45.6o

Lat.N 26.6oand Long. E.Two catalogues of earthquakes occurred on the territory of Romania were compiled, more or

less independently, by Radu (1974, 1980, 1995) and by Constantinescu and Marza (1980,1995) Table 5.1. The Radus catalogues are more complete, even the majority of significantevents are also included in the Constantinescu and Marza catalogue. The magnitude in Raducatalogue is the Gutenberg-Richter (1936) magnitude,MGR. The magnitude in Constantinescu& Marza catalogue was the surface magnitude, MS. Tacitly, that magnitude was laterassimilated as Gutenberg-Richter magnitude (Marza, 1995).

Table 5.1. Catalogue of subcrustal Vrancea earthquakes (Mw 6.3 ) occurred during the 20thcentury

RADU Catalogue,1994

MARZACatalogue,

1980

www.infp.ro

Catalogue,

1998

Date Time(GMT)

h:m:s

Lat. N Long.E

h,km I0 MGR Mw I0 Ms Mw

1903 13 Sept 08:02:7 45.7 26.6 >60 7 6.3 - 6.5 5.7 6.31904 6 Feb 02:49:00 45.7 26.6 75 6 5.7 - 6 6.3 6.6

1908 6 Oct 21:39:8 45.7 26.5 150 8 6.8 - 8 6.8 7.11912 25 May 18:01:7 45.7 27.2 80 7 6.0 - 7 6.4 6.7

1934 29 March 20:06:51 45.8 26.5 90 7 6.3 - 8 6.3 6.6 1939 5 Sept 06:02:00 45.9 26.7 120 6 5.3 - 6 6.1 6.2

1940 22 Oct 06:37:00 45.8 26.4 122 7 / 8 6.5 - 7 6.2 6.5

1940 10 Nov 01:39:07 45.8 26.7 1501) 9 7.4 - 9 7.4 7.7

1945 7 Sept 15:48:26 45.9 26.5 75 7 / 8 6.5 - 7.5 6.5 6.8

1945 9 Dec 06:08:45 45.7 26.8 80 7 6.0 - 7 6.2 6.51948 29 May 04:48:55 45.8 26.5 130 6 / 7 5.8 - 6.5 6.0 6.3

1977 4 March 2) 19:22:15 45.34 26.30 109 8 / 9 7.2 7.5 9 7.2 7.4

1986 30 Aug 21:28:37 45.53 26.47 133 8 7.0 7.2 - - 7.1

1990 30 May 10:40:06 45.82 26.90 91 8 6.7 7.0 - - 6.9

1990 31 May 00:17:49 45.83 26.89 79 7 6.1 6.4 - - 6.41)

Demetrescus original (1941) estimation: 150Km; Radus initial estimation(1974) was 133 km2) Main shock

Nov.10 , 1940destruction in the epicentral area and in Moldavia, around 1000 deaths, thelargest RC building in Bucharest (Carlton) collapsed;March 4, 1977 more than 1500 deaths, more than 11000 injured, more than 2 billion$ losses, 31 buildings with more than 4 stories collapsed.


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As a systematization requirement for seismic hazard assessment, usually it is recommendedthe use of the moment magnitude, Mw. For Vrancea subcrustal events the relation betweentwo magnitudes can be simply obtained from recent events data given in Table 5.1:

Mw MGR+ 0.3 6.0 < M GR< 7.7 (5.15)

Even the available catalogues of Vrancea events were prepared using the Gutenberg-RichtermagnitudeMGR,the recurrence-magnitude relationship was herein newly determined using themoment magnitude Mw. The relationship is determined from Radus 20

thcentury catalogue ofsubcrustal magnitudes with threshold lower magnitudeMw=6.3.The average number per year of Vrancea subcrustal earthquakes with magnitude equal to andgreater thanMw, as resulting also from Figure 5.5, is:

log n(Mw) = 3.76 - 0.73 Mw (5.16)

Figure 5.5. Magnitude recurrence relation for the subcrustal Vrancea source (Mw6.3)

The values of surface rupture area (SRA) and surface rupture length (SRL) from Wells andCoppersmith (1994) equations for "strike slip" rupture were used to estimate maximumcredible Vrancea magnitude. According to Romanian geologists Sandulescu & Dinu, in

Vrancea subduction zone: SRL 150200 km, SRAM)pery

r

log n (>Mw ) = 3.76 - 0.73Mw

20th

century Radu's catalogue

Mw, max= 8.17.8

6.3 6.7 7.1 7.5 7.9 8.3

)3.61.8(687.1

)1.8(687.1687.1654.8

1

1

e

eeMn

w

w

MM

w


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If the source magnitude is limited by an upper bound magnitude Mw,max, the recurrencerelationship can be modified in order to satisfy the property of a probability distribution,Equation 5.18:

0ww

ww

w

MM

MMM

w

e1

e1eMn

max,

max,

(5.18)

and, in the case of Vrancea source (Elnashai and Lungu 1995):

)..(.

).(...

36186871

M186871M68716548

we1

e1eMn

w

w

(5.19)

In Eq.(5.18), the threshold lower magnitude isMw0=6.3, the maximum credible magnitude of

the source isMw,max=8.1, and = 3.76 ln10 = 8.654, = 0.73 ln10 =1.687.The maximum credible magnitude of the source governs the prediction of source

magnitudes in the range of large recurrence intervals, where classical relationship (5.16) does

not apply, Table 5.3.

Table 5.3. Mean recurrence interval (MRI) of Vrancea magnitudes, (Mw)=1/n(Mw)

Date Gutenberg-Richter

Momentmagnitude,

MRIfrom Eq.(5.18),

MRIfrom Eq.(5.16),

magnitude,MGR Mw years years

8.1

8.0

7.9

-778

356

150127107

10 Nov. 1940 7.4

7.8

7.77.6

217

148108

91

7665

4 March 1977

30 Aug. 1986

7.2

7.0

7.5

7.4

7.3

7.2

8263

5040

55463733

30 May 1990 6.7 7.0 26 23

The depth of the Vrancea foci has a great influence on the experienced seismic intensity. Thedamage intensity of the Vrancea strong earthquakes is the combined result of both magnitudeand location of the focus inside the earth.

The relationship between the magnitude of a destructive Vrancea earthquake (Mw6.3)and thecorresponding focal depth shows that higher the magnitude, deeper the focus:

ln h = - 0.866 + 2.846 lnMw - 0.18 P (5.20)

whereP is a binary variable: P=0 for the mean relationship and P=1.0 for mean minus onestandard deviation relationship.The following model was selected for the analysis of attenuation (Mollas & Yamazaki, 1995):

ln PGA = c0+ c1Mw+ c2lnR +c3R +c4h + (5.21)

where: PGAis peak ground acceleration at the site,Mw- moment magnitude, R - hypocentraldistance to the site, h - focal depth, c0, c1, c2, c3, c4 - data dependent coefficients and - random

475

100

50


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variable with zero mean and standard deviation = ln PGA, Table 5.4. Details are givenelsewhere (Lungu et.al., 2000, Lungu et. al. 2001). To obtain the values of table 5.4 we used therecords from 1977, 1986 and 1990s earthquakes.

Table 5.4. Regression coefficients inferred for horizontal components of peak ground

acceleration during Vrancea subcrustal earthquakes, Equation (4.21)c0 c1 c2 c3 c4 lnPGA

3.098 1.053 -1.000 -0.0005 -0.006 0.502

The application of the attenuation relation 5.21 for the Vrancea subcrustal earthquakes ofMarch 4, 1977, August 30, 1986 and May 30, 1990 is represented in Figures 5.6, 5.7 and 5.8.

Attenuationrelation- March4, 1977;Mw=7.5,h=109km

0

50

100

150

200

250

300

350

400

450

500

0 50 100 150 200 250

D

, km

PGA,cm/s^2

median median+stdev

Figure 5.6. Attenuation relation applied for March 4, 1977 Vrancea subcrustal source

Attenuationrelation- August 30 1986;Mw=7.2, h=133km

0

50

100

150

200

250

0 50 100 150 200 250

D, km

PGA,cm/s^

median

median+stdev

Figure 5.7. Attenuation relation applied for August 30, 1986 Vrancea subcrustal source


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Attenuationrelation- May30, 1990;Mw=7.0, h=91km

0

50

100

150

200

250

300

350

400

0 50 100 150 200 250

D, km

PGA,cm/s^

median

median+stdev

Figure 5.8. Attenuation relation applied for May 30, 1990 Vrancea subcrustal source

For a given earthquake recurrence, the mean annual rate of exceedance of a particular value ofpeak ground acceleration, PGA*, is calculated using the total probability theorem (Cornell,1968, Kramer, 1996):

dmdrrfmfrmPGAPGAPPGAPGA RMM )()(),|*(*)( min (5.22)

where:

(PGA>PGA*)mean annual rate of exceedance ofPGA*;- Mminis the mean annual rate of earthquakes of magnitudeMlarger or equal thanMmin;- P(PGA>PGA*|m,r) probability of exceedance of PGA* given the occurrence of anearthquake of magnitude m at source to site distance r. This probability is obtained fromattenuation relationship (4.21) assuming log-normal distribution for PGA;-fM(m)probability density function for magnitude;-fR(r)probability density function for source to site distance.

The probability density function for magnitude is obtained from Eq. (4.8) (Kramer, 1996).The probability density function for source to site distance is considered, for the sake ofsimplicity, uniform over the rectangle of 40x80km2having the long axis oriented N45E andbeing centered at about 45.6oLat.N and 26.6oLong. E.

The mean annual rate of exceedance of PGA the hazard curve - for Bucharest site andVrancea seismic source is represented in Figure 5.9.

The hazard curve can be approximated by the form kgo akH , where ag is peak ground

acceleration, and ko and k are constants depending on the site (in this case ko=1.176E-05,k=3.0865).


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PSHA - Bucharest

1.E-04

1.E-03

1.E-02

1.E-01

100 150 200 250 300 350 400

PGA, cm/s2

Annualexceedancerate,

(PGA)

Figure 5.9. Hazard curve for Bucharest from Vrancea seismic source


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6. INTRODUCTION TO RANDOM (STOCHASTIC) PROCESSES THEORY

6.1. Background

Physical phenomena of common interest in engineering are usually measured in terms of

amplitude versus time function, referred to as a time history record. There are certain types of

physical phenomena where specific time history records of future measurements can be

predicted with reasonable accuracy based on ones knowledge of physics and/or prior

observations of experimental results, for example, the force generated by an unbalanced

rotating wheel, the position of a satellite in orbit about the earth, the response of a structure to

a step load. Such phenomena are referred to as deterministic, and methods for analyzing their

time history records are well kn

structural reliability and risk analysis courses

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