structural reliability and risk analysis book

Technical University of Civil Engineering of Bucharest

Reinforced Concrete Department

STRUCTURAL RELIABILITY AND RISK ANALYSIS

Lecture notes

STRUCTURAL RELIABILITY AND RISK ANALYSIS – 4th Year FILS

UTCB, Technical University of Civil Engineering, Bucharest 2

Foreword

These lecture notes provides an insight into the concepts, methods and procedures ofstructural reliability and risk analysis considering the presence of random uncertainties. Thecourse is addressed to undergraduate students from Faculty of Engineering in ForeignLanguages instructed in English language as well as to postgraduate students in structuralengineering. The objectives of the courses are:

- to provide a review of mathematical tools for quantifying uncertainties using theoriesof probability, random variables and random processes;

- to develop the theory of methods of structural reliability based on concept of reliabilityindices;

- to explain the basics of code calibration;

- to evaluate actions on buildings and structures due to natural hazards;

- to provide the basics of risk analysis;

- to provide the necessary background to carry out reliability based design and risk-based decision making and to apply the concepts and methods of performance-basedengineering;

- to prepare the ground for students to undertake research in this field.

The content of the lectures in Structural Reliability and Risk Analysis is:

- Introduction to probability and random variables; distributions of probability

- Formulation of reliability concepts for structural components; exact solutions, first-order reliability methods; reliability indices; basis for probabilistic design codes

- Seismic hazard analysis

- Evaluation of snow, wind and earthquake loads for design of structures.



Table of Contents

1. INTRODUCTION TO RANDOM VARIABLES THEORY................................................ 61.1 Data samples .................................................................................................................... 71.2 Indicators of the sample (esantion) .................................................................................. 71.3 Probability ........................................................................................................................ 91.4 Random variables ........................................................................................................... 101.5 Indicators of the probability distributions ...................................................................... 12

2. DISTRIBUTIONS OF PROBABILITY .............................................................................. 152.1. Normal distribution ....................................................................................................... 152.2. Log-normal distribution ................................................................................................ 192.3. Extreme value distributions........................................................................................... 21

2.3.1. Gumbel distribution for maxima in 1 year ............................................................. 222.3.2. Gumbel distribution for maxima in N years........................................................... 25

2.4. Mean recurrence interval............................................................................................... 273. FUNCTION OF RANDOM VARIABLES ......................................................................... 29

3.1 Second order moment models ........................................................................................ 294. STRUCTURAL RELIABILITY ANALYSIS ..................................................................... 33

4.1. The basic reliability problem......................................................................................... 334.2. Special case: normal random variables ......................................................................... 354.3. Special case: log-normal random variables................................................................... 364.4. Calibration of partial safety coefficients ....................................................................... 37

5. SEISMIC HAZARD ANALYSIS........................................................................................ 415.1. Deterministic seismic hazard analysis (DSHA) ............................................................ 415.2. Probabilistic seismic hazard analysis (PSHA) .............................................................. 425.3. Earthquake source characterization............................................................................... 435.4. Predictive relationships (attenuation relations) ............................................................. 465.5. Temporal uncertainty .................................................................................................... 465.6. Probability computations............................................................................................... 475.7. Probabilistic seismic hazard assessment for Bucharest from Vrancea seismic source . 47

6. ACTIONS ON STRUCTURES. SNOW LOADS ............................................................... 536.1 Introduction .................................................................................................................... 536.2 Snow load on the ground................................................................................................ 536.3 Snow load on the roof .................................................................................................... 566.4 Roof shape coefficients .................................................................................................. 586.5 Local effects ................................................................................................................... 60

6.5.1 Local verification .................................................................................................... 606.5.2 Exceptional snow drift on the roof .......................................................................... 61

7. ACTIONS ON STRUCTURES. WIND ACTION .............................................................. 627.1. General .......................................................................................................................... 627.2 Reference wind velocity and reference velocity pressure .............................................. 637.3 Terrain roughness and Variation of the mean wind with height .................................... 647.4 Wind turbulence ............................................................................................................. 677.5 Peak values ..................................................................................................................... 677.6 Wind actions................................................................................................................... 68

7.6.1 Wind pressure on surfaces....................................................................................... 697.6.2 Wind forces ............................................................................................................ 70

8. ACTIONS ON STRUCTURES. SEISMIC ACTION ......................................................... 718.1 Introduction .................................................................................................................... 718.2 Representation of the seismic action.............................................................................. 71



8.3 Design spectrum for elastic analysis .............................................................................. 74

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)

3. “EN 1991 – Eurocode 1: Actions on structures. Part 1-3 General actions – Snow Loads”4. EN 1991-1-4 - Eurocode 1: Actions on structures - Part 1-4: General actions - Windactions, CEN

5. CR 1-1-4/2012„Cod de proiectare. Evaluarea acţiunii vântului asupra construcţiilor”.6. CR 1-1-3/2012 „Cod de proiectare. Evaluarea acţiunii zăpezii asupra construcţiilor”.7. P100-1/2013 „Cod de proiectare seismica - Partea I – Prevederi de proiectare pentruclădiri”.



Examination subjects:

1. Random variables, indicators of the sample, histograms2. Probability, probability distributions function, cumulative density function, indicators3. 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. Snow loads11. Wind action12. Seismic action



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 reliability

phenomena and informations that cannot be reduced through better knowledge.- Epistemic uncertainties – that are related to our inability to predict the future

behavior 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 than – see the class of concrete)Structural Analysis (3) - Epistemic uncertainty (linear elastic materials – strains proportionalto stresses)Sectional Analysis (4) - Random + Epistemic uncertaintyDesign for some limits states (5) – ultimate limit states (safety), serviceability limit states.For power plants – LOCA – local lose of cooling agent

Uncertainties can be reduced to physical models:- random variable (RV) – dead load, material strength- random process – wind 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 evaluatingexperiments 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



the variation due 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 (thepopulation).

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 thatthe 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, …, xn or, briefly, sample mean, is denoted by_

x (or mx) andis defined by the formula:

n

jjx

nx

1

_ 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 average is used for_

x .

The variance (dispersion) of a sample x1, x2, …, xn or, briefly, sample variance, is denoted bysx

2 and is defined by the formula:

n

jjx xx

ns

1

2_

2 )(1

1(1.2)

The sample variance is 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 s2 is called the standard deviation of the sample and is

denoted by sx.2xx ss . The mean, mx and the standard deviation, sx has the same units.

The coefficient of variation of a sample x1, x2, …,xn is denoted by COV and is defined as theratio of the standard deviation of the sample to the sample mean

xsCOV (dimensionless) (1.3)



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 data in 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 0.400360 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 absolutefrequency of that value x 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 that x. 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 relativefrequencies

Figure 1.2 Histogram of cumulativerelative frequencies



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 of groupingmay 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 called class 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 -

nn

f jj and

m

jjf

1

= 1). The normalized relative frequency isx

ff jN

j . The normalization is

with respect to the x.

The relative frequency is called the frequency function of the grouped sample, and thecorresponding cumulative relative class frequency is 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 value xj coincides with the common point of two class intervals, one takes itinto the class interval that extends from xj to the right.

1.3 ProbabilityProbability 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. If x 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



• 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. Theset 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 of S.

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 E in suchan experiment there is a number P(E) such that the relative frequency of E in 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 called probability of anevent E in that random experiment. Note that this number is not an absolute property of E butrefers 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 X corresponding to that experiment has assumedthe value a. The corresponding probability is denoted by P(X=a).Similarly, the probability of the event X assumes any value in the interval a<X<b is denotedby P(a<X<b).The probability of the event X≤ c (X assumes any value smaller than c or equal to c) isdenoted by P(X≤ c), and the probability of the event X>c (X assumes any value greater than c)is denoted by P(X>c).The last two events are mutually exclusive:

P(X≤c) + P(X>c) = P(-∞ < X < ∞) =1 (1.4)

The random variables are either discrete or continuous.

dxxfdxxXxP x )()( - Definition of probability density function (PDF)

b

a

duufbXaP )()( (1.5)

Hence this probability equals the area under the curve of the density f(x) between x=a andx=b, as shown in Figure 1.3.



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

f(x)

ba

P(a<X<b)

Figure 1.3 Example of probability computation

Properties of PDF:- fx(x) >0- As x - fx(x) 0- As x fx(x) 0

- 1du)u(f

(the most important property) (1.6)

If X is any random variable, then for any real number x there exists the probability P(X ≤ x)corresponding to X ≤ x (X assumes any value smaller than x or equal to x) that is a function ofx, which is called the cumulative distribution function of X, CDF and is denoted by F(x).Thus F(x) = P(X ≤ x).Since for any a and b > a one has P(a ≤ X <b) = P(X ≤ b) - P(X ≤ a)

One shall now define and consider continuous random variables. A random variable X and thecorresponding distribution are said to be of continuous type or, briefly, continuous if thecorresponding distribution function F(x) = P(X x) can be represented by an integral in theform

x

du)u(f)x(F cumulative distribution function (CDF) (1.7)

where the integrand is continuous and is nonnegative. The integrand f is called the probabilitydensity or, briefly, the density of the distribution. Differentiating one notices that

F’(x) = f(x)In this sense the density is the derivative of the distribution function.

Properties of CDF:

Fx(x)

x

Fx(a)

0a b

Shape ofCDF



- 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 <a); Fx(b) = P(X <b) one has:

)a(F)b(Fdu)u(f)bXa(Pb

a

1.5 Indicators of the probability distributions

- Refers to the population -The mean value or mean of a distribution is denoted by and is defined by

j

jj )x(fx (discrete distribution) (1.8a)

dx)x(xf (continuous distribution) (1.8b)

where f(xj) is the probability function of discrete random variable X and f(x) is the density ofcontinuos random variable X.The mean is also known as the mathematical expectation of X and is sometimes denoted byE(X).

The variance of a distribution is denoted by 2x and is defined by the formula:

(1.9)The positive square root of the variance is called the standard deviation and is denoted by x.Roughly speaking, the variance is a measure of the spread or dispersion of the values, whichthe corresponding random variable X can assume.

The coefficient of variation of a distribution is denoted by Vx and is defined by the formula

x

xxV

(1.10)

The central moment of order i of a distribution is defined as:

(1.11)

μ2 = 2x the variance is the central moment of second order.

The skewness coefficient is defined with the following relation:

(1.12)Skewness is a measure of the asymmetry of the probability distribution of a random variable.

A distribution is said to be symmetric with respect to a number x = c if for every real x,f(c+x) = f(c-x). (1.13).



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 thepeak 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(X ≤ xp) = p).



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 variable X has mean μX and variance 2x , then the corresponding variable

Z = (X - μX)/σX has the mean 0 and the variance 1. Z is called the standardized variablecorresponding to X.

)(xFX

xxpC

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)x(F pX

1

)(xFX

xxpCum

ulat

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dist

ribu

tion

func

tion

p

)(xfx

xxpPro

babi

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dis

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utio

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xxpP

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2. DISTRIBUTIONS OF PROBABILITY

2.1. Normal distribution

The continuous distribution having the probability density function, PDF

2

2

11

2

1)(

X

Xx

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. Thisdistribution is very important, because many random variables of practical interest are normalor 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 of

f(x) is called the bell-shaped curve. It is symmetric with respect to and is biparametric (μX

and 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=0.10

V=0.20

V=0.30

Figure 2.1. PDF’s of the normal distribution for various values of V

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 PDF’s areat left and right of x = x =



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 600x

F(x)

V=0.10

V=0.20

V=0.30

Figure 2.2. CDF’s of the normal distribution for various values of V

From (2.2) one obtains

dvaFbFbXaPb

a

v

X

e X

X

2

2

11

2

1)()()(

(2.3)

In Microsoft Excel: NORMDIST (x, μX and x ) – normal distributionChange of variable:

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

In fact, if one sets uv

x

x

, then

xdv

du

1

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

integral (2.2) from - to z =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:



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 where z =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 normaldistribution

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 normaldistribution

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. PDF and CDF of 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, μz=0 and z = 1) – normal standard distribution

2

1



From (2.3) and (2.5) one gets:

X

X

X

X abaFbFbXaP

)()()( (2.6).

In particular, when a = XX and b = XX , the right-hand side equals (1) - (-1);

to a = XX and b = XX there corresponds the value (2) - (-2), etc. Using

tabulated values of function 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 variable Xwill 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 99¾ % 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 X with p non-exceedanceprobability ( P(X xp) = p=Fx (xp)) is computed as follows:

xp = X + kpX (2.8).

The meaning of kp becomes clear if one refers to the reduced standard variable z = (x - .

Thus, x = +z and kp represents the value of the reduced standard variable for which:

(z) = p.

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

p < 0,5 kp <0p > 0,5 kp >0p = 0,5 kp =0 – we define the median valueThe most common values of kp are given in Table 2.1.

Table 2.1. Values of kp for different non-exceedance probabilities p (from Tables)

p 0.01 0.02 0.05 0.95 0.98 0.99kp -2.326 -2.054 -1.645 1.645 2.054 2.326



2.2. Log-normal distribution

The lognormal distribution of random variable x is the normal distribution of the randomvariable lnx.

The log-normal distribution (Hahn & Shapiro, 1967) is defined by its following property: ifthe 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 distributionfunction CDF of random variable lnX is 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

Xln2

Xln

Xln

(2.9).

Since:

x

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

the probability density function PDF results 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 thedistribution is shifted to the left. The skewness coefficient for lognormal distribution is:

31 3 XX VV (2.12)

where VX is the coefficient of variation of random variable X. Higher the variability, higherthe shift of the lognormal distribution.

The mean and the standard deviation of the random variable lnX are related to the mean andthe standard deviation of the random variable X as follows:

21 X

XXln

Vln

(2.13)

)Vln( XXln21 (2.14).

In the case of the lognormal distribution the following relation between the mean and themedian holds true:

50.Xln xln (2.15).

Combining (2.13) and (2.15) it follows that the median of the lognormal distributions islinked to the mean value by the following relation:

250

1 X

X.

Vx

(2.16)



If VX is small enough (VX 0.1), 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 of V

The mode ≠ median ≠ mean of the distribution.



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 - to z =v

vv

ln

lnln

. From (2.9) one obtains:

dueduvv

ezz u

v

v u

v

vv

2

ln

/)(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 X with p non-exceedanceprobability (P(X xp) = p) is computed as follows, given lnX normally distributed:

ln(xp) = lnX + kplnX (2.20)

From (2.20) one gets:

XpX kp ex lnln (2.21)

where kp represents 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

X abaFbFbXaP

)()()(

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 (1902–1985). EmilGumbel codified the theory of extreme values in his book “Statistics of extremes” publishedin 1958 at Columbia University Press.

Extreme value distributions are the limiting distributions for the minimum or the maximum ofa very large collection of random observations from the same arbitrary distribution. Theextreme value distributions are of interest especially when one deals with natural hazards likesnow, wind, temperature, floods, etc. In all the previously mentioned cases one is notinterested in the distribution of all values but in the distribution of extreme values, whichmight be the minimum or the maximum values. In Figure 2.5 it is represented the distributionof all values of the random variable X as well as the distribution of minima and maxima of X.

Year 1: y11, y21, y31,… ym1 - x1

Year 1: y12, y22, y32,… ym2 - x2

…..Year n: y1n, y2n, y3n,… ymn - xn

Xi = max (y1i, y2i, y3i,… ymi) –maximum annual value



0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0 100 200 300 400 500 600 700 800 900 1000x

f(x)

all valuesdistribution

minimadistribution

maximadistribution

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 u and are the parameters:

u = x –

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

n

knnln

klim

1

1

x

1

6

The final form is: u = x – 0.45x – mode of the distribution (Figure 2.8) and

= 1.282 / x – dispersion coefficient (shape factor).

The skewness coefficient of Gumbel distribution is positive constant ( 139.11 ), i.e. the

distribution is shifted to the left. In Figure 2.6 it is represented the CDF of Gumbeldistribution for maxima for the random variable X with the same mean x and differentcoefficients of variation Vx.The probability distribution function, PDF is obtained straightforward from (2.22):

)ux(e)ux( eedx

)x(dF)x(f

(2.23)

The PDF of Gumbel distribution for maxima for the random variable X with the same mean

x and different coefficients of variation Vx is represented in Figure 2.7.



Gumbel distributionfor 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=0.20

V=0.30

Figure 2.6. CDF of Gumbel distribution for maxima for the random variable Xwith the same mean x and different coefficients of variation Vx

Gumbel distributionfor 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. PDF of Gumbel distribution for maxima for the random variable Xwith the same mean x and 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.



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

f(x)

0.45 x

u x

Figure 2.8. Parameter u in Gumbel distribution for maxima

Fractile xp:

Given X follows Gumbel distribution for maxima, the fractile xp that is defined as the value ofthe 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

xGpx

xxxp k)plnln(

282.145.0)plnln(

1ux

(2.25)

where:

)plnln(78.045.0k Gp (2.26).

The values of kpG for different non-exceedance probabilities are given in Table 2.2.

Table 2.2. Values of kpG for different non-exceedance probabilities p

p 0.50 0.90 0.95 0.98kp

G -0.164 1.305 1.866 2.593

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

xm

0,164 σx



2.3.2. Gumbel distribution for maxima in N years

All the previous developments are valid for the distribution of maximum yearly values. If oneconsiders the probability distribution in N (N>1) years, the following relation holds true (ifone considers that the occurrences of maxima are independent events):

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

F(x)N years = P(X x) in N years= P(X x)year1 ∙ P(X x)year2 ∙…∙ P(X x)year N

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

where:

F(x)N years – CDF of random variable X in N years

F(x)1 year – CDF of random variable X in 1 year.

The Gumbel distribution for maxima has a very important property – the reproducibility ofGumbel distribution - i.e., if the annual maxima (in 1 year) follow a Gumbel distribution formaxima then the maxima in N years will also follow a Gumbel distribution for maxima:

)ux()ux( NeN

eNN ee)x(F)x(F

11111

)Nux(N))

Nlnu(x(

Nln)ux( eee eee

1

1111

(2.28)where:u1 – mode of the distribution in 1 year– dispersion coefficient in 1 year

uN = u1 + lnN / – mode of the distribution in N years

– dispersion coefficient in N yearsThe PDF of Gumbel distribution for maxima in N years is translated to the right with the amount

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

Gumbel distributionfor maxima

0

0.0021

0.0042

0.0063

100 200 300 400 500 600 700 800 900 1000x

f(x)

N yr.

1 yr.

u 1 u N

lnN / 1

m 1 m N

lnN / 1

Figure 2.9. PDF of Gumbel distribution for maxima in 1 year and in N years



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 theamount lnN / with respect to the CDF of Gumbel distribution for maxima in 1 year, Figure 2.10.


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

100 200 300 400 500 600 700 800 900 1000x

F(x)

N yr.

1 yr.

Figure 2.10. CDF of Gumbel distribution for maxima in 1 year and in N years

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 if N is large,N 50) if Gumbel distribution for maxima in N years is employed, Figure 2.11.


0

0.0021

0.0042

0.0063

100 200 300 400 500 600 700 800 900 1000x

f(x)

N yr.

1 yr.

probability of

exceedance of x p

in N years

probability of

exceedance of x p

in 1 year

x p

Figure 2.11. Superior fractile xp in 1 year and its significance in N year



2.4. Mean recurrence interval

The loads due to natural hazards such as earthquakes, winds, waves, floods were recognizedas having randomness in time as well as in space. The randomness in time was considered interms of the return period or recurrence interval. The recurrence interval also known as areturn 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, floodor river discharge flow of a certain intensity or size. It is a statistical measurement denotingthe average recurrence interval over an extended period of time. The actual time T betweenevents is a random variable.

The mean recurrence interval, MRI of a value larger than x of the random variable X may bedefined as follows:

pxFxXPxXP)xX(MRI

xyearyear

1

1

1

1

1

11

11(2.29)

where:p is the annual probability of the event (X≤x) and FX(x) is the cumulative distribution function of X.

Thus the mean recurrence interval of a value x is equal to the reciprocal of the annualprobability of exceedance of the value x. The mean recurrence interval or return period has aninverse 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 anda 50-year flood has a 0.02 (2%) chance of being exceeded in any one year. It is commonlyassumed 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 bestatistically true over thousands of years, it is incorrect to think of the return period in thisway. The term return period is actually misleading. It does not necessarily mean that thedesign earthquake of a 10 year return period will return every 10 years. It could, in fact, neveroccur, or occur twice. This is why the term return period is gradually replaced by the termrecurrence interval. Researchers proposed to use the term return period in relation with theeffects and to use the term recurrence interval in relation with the causes.

The mean recurrence interval is often related with the exceedance probability in N years. Therelation among MRI, N and the exceedance probability in N years, Pexc,N is:

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

RP – return period

Values of a randomvariable (natural hazard)

x

years

RP it is not the same(random variable)



Pexceedance, N (>x) = 1 - TNN

exT

1)(

11 (2.30)

pxXPxXMRI

yearp

1

11

1)(

1

; )( pxXMRI = )( pxXT ;

example: x0.98 similar50Tx ; yearsMRI 50

98,011

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 consideringN=50 years.

Table 2.3 Correspondence amongst MRI, Pexc,1 year and Pexc,50 years

Mean recurrence interval,

years MRI, )( xT Probability of exceedance

in 1 year, Pexc,1 year

Probability of exceedancein 50 years, Pexc,50 years

10 0.10 0.9930 0.03 0.8150 0.02 0.63100 0.01 0.39225 0.004 0.20475 0.002 0.10975 0.001 0.052475 0.0004 0.02

The modern earthquake resistant design codes consider the definition of the seismic hazardlevel based on the probability of exceedance in 50 years. The seismic hazard due to groundshaking is defined as horizontal peak ground acceleration, elastic acceleration responsespectra or acceleration time-histories. The level of seismic hazard is expressed by the meanrecurrence interval (mean return period) of the design horizontal peak ground acceleration or,alternatively by the probability of exceedance of the design horizontal peak groundacceleration 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 Table2.4. The correspondence between the mean recurrence interval and the probability ofexceedance 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 exceedancein 50 years of design horizontal peak ground acceleration as in FEMA 356

Seismic HazardLevel

Mean recurrence interval(years)

Probability ofexceedance

SHL1SHL2SHL3SHL4

722254752475

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



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, W are described deterministically, the design condition (ultimate limit state condition)is:

capMM max ; Wql

y 8

2

and considering

S – sectional effect of load, and

R – sectional resistance

it follows that:

8

2qlS ; WR y .

The following question rises:

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

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

xfx , )(2

xfx … )(xfnx ;

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 q and S ( y and R) is linear

2. The relation is non linear.

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

bXaYX (a and b constants, X random variable)



For the random variable X one knows: the probability density function, PDF, the cumulativedistribution function, CDF, the mean, m and the standard deviation, . The unknowns are thePDF, CDF, m and for the random variable Y.

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

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

a

byx

;

a

byf

axf

adx

dyxfyf XXXY

1)(

11)()( (3.2)

Figure 3.2. Linear relation between random variables X and Y

Distribution of Y Distribution 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

dxxf X )(

dyyfY )(

)(xf X

?)( yfY

x

y



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

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

find fy, ym , y

Observations:

1. Let the random variable nXXXXY 321 . If the random variables Xi

are normally distributed then Y is also normally distributed.

2. Let the random variable XXXXY 321 . If the random variables Xi are

log-normally distributed then Y is also log-normally distributed.

Let the random variables Xi with known mean and standard deviations:

nim

Xi

i

X

X

i ,1;

....)(!2

)()(

!1)(

)()( 2

axafaxafafxf Taylor series

The first order approximation

FORM – First order reliability method

If one develops the function Y in Taylor series around the mean m =(1xm ,

2xm …ixm ..

nxm ),

and keeps only the first order term of the series, then the mean and the standard deviation ofthe 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)

2y 2

2

1iX

m

n

i iXY

- First order approximation of the variance

n

i iXminX

mnX

mX

mY x

y

x

y

x

y

x

y

1

22

22

22

2

2

21

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 of FOSMM are 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



bXaY bmam XY

222XY a ; XY a

21 XXY

Ym Y (1xm ,

2xm ) =21 XX mm

2

2

2

1

2

1

2

2

2

2

22

2

1

21

2

2

2

2

2

1

2

1

2XXXXXmXmX

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

2XX

X

X

X

X

XX

XXXX

Y

YY VV

mmmm

mm

mV



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 that R and S are 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(ln R-ln S0) (4.1d)

or, in general

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

where G( ) is termed the limit state function and the probability of failure is identical with theprobability of limit state violation.

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

Pf + Re=1

(R>S – safe; R=S – limit state; R<S – failure)

Quite general density functions fR and fS for R and S respectively are shown in Figure 4.1together with the joint (bivariate) density function fRS(r,s). For any infinitesimal element(rs) the latter represents the probability that R takes on a value between r and r+r and S a

value between s and s+s as r and s each approach zero.

Figure 4.1. Joint density function fRS(r,s), marginal density functions fR(r) and fS(s)and failure domain D, (Melchers, 1999)



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).

When R and S are independent, fRS(r,s)=fR(r)fS(s) and relation (3.2) becomes:

dxxfxFdsdrsfrfSRPP SR

rs

SRf 0 (4.3)

Relation (4.3) is also known as a convolution integral with meaning easily explained byreference to Figure 4.2. FR(x) is the probability that Rx or the probability that the actualresistance R of the member is less than some value x. Let this represent failure. The term fs(x)represents the probability that the load effect S acting in the member has a value between xand x+x as x0. By considering all possible values of x, 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 densityfunctions fR(r) and fS(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

-4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11x

F R (x), f S (x)

R

S

R=x

P(R<x)

Figure 4.2. Basic R-S problem: FR( ) fS( ) representation

0

0.0021

0.0042

0.0063

100 200 300 400 500 600 700 800 900 1000x

f R (x), f S (x)

R

S

amount of overlapof fR( ) and fS() –

rough indicator of pf

Figure 4.3. Basic R-S problem: fR( ) fS( ) representation

μS Sk Rk μR



If μS is distanced from μR we deal with Pf small.If μS is near to μR we deal with Pf large.

An alternative to expression (4.3) is:

dxxfxFP RSf )()(1 (4.4)

which is simply the sum of the failure probabilities over all cases of resistance for which theload effect exceeds the resistance.

4.2. Special case: normal random variables

For a few distributions of R 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 S andvariances R

2 and S2 respectively.

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

Z = R - S (4.5a)

Z2 = R

2 + 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

f Z (z) z

z0

P f

Z<0 Z>0

Failure Safety

Figure 4.4. Distribution of safety margin Z = R – S

0 r

s

Failure domain(z<0)

Safe domain(z>0)

z = r - s = 0Limit statefunction



The random variable Z = R - S is shown in Figure 4.4, in which the failure region Z 0 isshown shaded. Using (4.5) and (4.6) it follows that (Cornell, 1969):

1

022SR

SRfP (4.7)

where =z

z

is defined as reliability (safety) index.

If either of the standard deviations R and S or both are increased, the term in square bracketsin (4.7) will become smaller and hence Pf will increase. Similarly, if the difference between themean of the load effect and the mean of the resistance is reduced, Pf increases. Theseobservations may be deduced also from Figure 4.3, taking the amount of overlap of fR( ) and fS()as a rough indicator of Pf.

Pf β10-3 3.0910-4 3.7210-5 4.2710-6 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 followingparameters: means lnR and lnS and variances lnR

2 and lnS2 respectively.

The safety margin Z=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( 2

lnRelation (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 region Z 0 is shown

shaded.

Using (4.8) and (4.9) it follows that



1

0

22SR

S

R

fVV

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

f Z (z) z

z0

P f

Z<0 Z>0

Failure Safety

Figure 4.5. Distribution of safety margin Z=S

Rln

4.4. Calibration of partial safety coefficients

In USA: LRDF – Load and resistance factor design

22

ln

SR

S

R

VV

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

331

S

R

V

V . Given Lindt’s linearization it follows that:



SR

S

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 VS

VR ee (4.13)

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

Rq and Sp are characteristics values (fractiles) of the random variables R respectively S.

RqRqq VkR

VkRRkRq eeeeR lnlnln

SpSpp VkS

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

SVkp

q

RVkq

pq eSeR

)()(

RVkq

qe )( - Partial safety coefficient for sectional resistance

SVkp

pe )( - Partial safety coefficient for the effects of the load

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

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

designdesign

R

R

S

S 05.005.098.098.0 (4.14)

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

Assuming that S and R are log-normally distributed, one has:

RR VR

VRRR eeeeR 645.1645.1lnln645.1ln05.0 (4.15)

SS VS

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)



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 on

the 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 increases

while the partial safety coefficient for resistance 05.0 decreases. The theory of partial safety

coefficients 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

g0.98

VS0.6

0.7

0.8

0.9

0.05 0.1 0.15 0.2

F0.05

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)= 4,7 – ultimate Limit states (ULS)= 2,9 – Serviceability Limit states (SLS)

Actions (F) : Permanent actions (G), Variable actions (Q), Accidental action (A)Fk – characteristic value (upper fractile)Fd – design value

f - partial safety coefficient applied to the action

Fd = f · Fk

Ed = Sd · E(Fd) - design value of section effort of the load

E(Fd) – sectional effect of the design action (load)

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

sectional effort of the load



Ed = E( Sd · f · Fk) = E( F · Fk)

Xk – characteristic value of the strength of the material (inferior fractile)Xd – design value of the strength of the material

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

Xd = km

X1

Rd – design 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, Figure 4.7 are:

the combination value, represented as a product of ψ0Qk used for the verification ofultimate limit states and irreversible serviceability limit states;

the frequent value, represented as a product ψ1Qk, 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 ψ2Qk, (ψ2 ≤ 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

0 10 20 30 40 50 60 70

Timp

Valoareinstantanee Q

Valoare caracteristica, Qk

Valoare de combinatie, Ψ0Qk

Valoare frecventa, Ψ1Qk

Valoare cvasipermanenta, Ψ2Qk

0

0.01

0.02

0.03

0.04

05

1015

2025

f Q

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

iEdj,k

n

jQAPG 2

11

- seismic combination



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 400PGA , cm/s2

Ann

ual e

xcee

danc

e ra

te, l

(PG

A)

Hazard curve for Bucharest from Vrancea seismic source

y

yMRI1

)( ; λy – mean 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;

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

3. Selection of controlling earthquake (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;

M- Source (focus)

R – hypocentraldistance

h – depth

Epicenter Site - epicentraldistance



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 in DSHA

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 of DSHA, it is necessary to obtain the probability distribution ofpotential rupture location within the source and the probability distribution of source–to-site distance (longitude, latitude, magnitude and depth – are random variable);

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;

Site

Source 1M1

Source 3M3

Source 2M2

R1

R3

R2

Step 1 Step 2

Distance

GroundMotionParameter,Y

R2 R3 R1

M2M1

M3

Step 3 Step 4



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 sources – if the source is a short shallow fault area sources – if the source is a long and/or deep fault volumetric sources.

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 thetemporal occurrence of earthquakes. The recurrence law gives the distribution of earthquakesizes in a given period of time.

Site

Source 1M1

Source 3M3

Source 2M2

Step1

Step2

Step3

Step4

Magnitude

Averagerate,log

2

3

1

Distance, R

GroundMotion

Parameter,Y

P(Y>y*)

Parameter value, y*

R

f

R

f

R

f



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 inGutenberg & Richter’s work was the mean annual rate of exceedance, M of an earthquake ofmagnitude M which is equal to the number of exceedances of magnitude M divided 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 magnitude M,M - magnitude,a and b – numerical coefficients depending on the data set.

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

NM

The physical meaning of a and b coefficients can be explained as follows:0 =10a – mean annual number of earthquakes of magnitude greater than or equal to 0 (agives the global seismicity)

b – describes the relative likelihood of large to small earthquakes. If b increases thenumber of larger magnitude earthquakes decreases compared to those of smaller earthquakes(b is the slope of the recurrence plot).

Figure 5.3. The Gutenberg-Richter law

The a and b coefficients 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 PSHA is intended toevaluate the hazard form discrete, independent releases of seismic energy.

lg M =10ln

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

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

M = Me (5.4)

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

0 M

lg M10a

b

b small

b large

1



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 Mmin are eliminated,one gets (McGuire and Arabasz, 1990):

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

=minM

M1

= 1 -

minM

M

e

e

= 1 - )MM( mine (5.5)

fM(M) =dM

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

Mmin is the mean annual rate of earthquakes of magnitude M larger or equal than Mmin.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 for M = Mmax. This yields:

)(MFM = P[Mag. M Mmin M Mmax] =)M(F

)M(F

maxM

M =)MM(

)MM(

minmax

min

e1

e1

(5.7)

)(Mf M =dM

MdFM )(=

)MM(

)MM(

minmax

min

e1

e

. (5.8)

The mean annual rate of exceedance of an earthquake of magnitude M is:

)](1[min

MFMMM =)MM(

)MM()MM(

Mminmax

minmaxmin

mine1

ee

(5.9)

whereminM = minMe is the mean annual rate of earthquakes of magnitude M larger or

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

M = minMe )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 = Me )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 Y is a ground motionparameter, M is the magnitude of the earthquake, R is the source-to-site distance and Pi areother parameters taking into account the earthquake source, wave propagation path and siteconditions. The predictive relationships are used to determine the value of a ground motionparameter 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 Y exceeds acertain value, y* for an earthquake of magnitude, m at a given distance r (Figure 5.4):

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

Y

R

y*

P(Y>y*|m,r)

r

fY(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:

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 isnegligible.

If N is the number of occurrences of a particular event during a given time interval, theprobability of having n occurrences 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 PSHA are given as seismic hazard curves quantifying the annual probabilityof exceedance of different values of selected ground motion parameter.The probability of exceedance a particular value, y* of a ground motion parameter (GMP) Y iscalculated 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)

where X is a vector of random variables that influence Y (usually magnitude, M and 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 of y* given the occurrence of an earthquake ofmagnitude m at 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 + c1 Mw + c2 lnR +c3R +c4 h +

where: PGA is peak ground acceleration at the site,Mw- moment magnitude,R - hypocentral distance to the site,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 20th century seismicity, the epicentral Vrancea area is confined to arectangle of 40x80km2 having the long axis oriented N45E and being centered at about 45.6o

Lat.N 26.6o and Long. E.

M- Source (focus)

R – hypocentraldistance

h – depth

Epicenter Site - epicentraldistance



Two catalogues of earthquakes occurred on the territory of Romania were compiled, more orless independently, by Radu (1974, 1980, 1995) and by Constantinescu and Marza (1980,1995) Table 5.1. The Radu’s 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 20th century

Date Time(GMT)h:m:s

Lat. N Long.E

RADU Catalogue,1994

MARZACatalogue,

1980

www.infp.roCatalogue,

1998h, 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 6.71934 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 6.51940 10 Nov 01:39:07 45.8 26.7 1501) 9 7.4 - 9 7.4 7.71945 7 Sept 15:48:26 45.9 26.5 75 7 / 8 6.5 - 7.5 6.5 6.81945 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 6.31977 4 March 2) 19:22:15 45.34 26.30 109 8 / 9 7.2 7.5 9 7.2 7.41986 30 Aug 21:28:37 45.53 26.47 133 8 7.0 7.2 - - 7.11990 30 May 10:40:06 45.82 26.90 91 8 6.7 7.0 - - 6.91990 31 May 00:17:49 45.83 26.89 79 7 6.1 6.4 - - 6.4

1) Demetrescu’s original (1941) estimation: 150Km; Radu’s initial estimation (1974) was 133 km2) Main shock

Nov.10 , 1940 – destruction 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.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 < MGR < 7.7 (5.15)

Even the available catalogues of Vrancea events were prepared using the Gutenberg-Richtermagnitude MGR, the recurrence-magnitude relationship was herein newly determined using themoment magnitude Mw. The relationship is determined from Radu’s 20th century catalogue ofsubcrustal magnitudes with threshold lower magnitude Mw=6.3.The average number per year of Vrancea subcrustal earthquakes with magnitude equal to andgreater than Mw, 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, inVrancea subduction zone: SRL 150200 km, SRA<8000 km2. Based on this estimation, fromTable 5.2 one gets:

Mw,max= 8.1. (5.17)

Table 5.2. Application of Wells and Coppersmith equations to the Vrancea source (mean values)

M Mw Event Experienced Wells & Coppersmith equationsSRA, km2 logSRA=-3.42+0.90Mw logSRL=-3.55+0.74Mw

SRA, km2 SRL, km6.7 7.0 May 30,1990 11001) 759 437.0 7.2 Aug. 30,1986 14001) 1148 607.2 7.5 March 4, 1977 63 x 37=23312) 2138 100

8.1 Max. credible - 7413 2781)As cited in Tavera (1991) 2) Enescu et al. (1982)

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

we1

e1eMn

max,

max,

(5.18)

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

)..(.

).(...

36186871

M186871M68716548

we1

e1eMn

ww

(5.19)

0.001

0.01

0.1

1

6.0 6.4 6.8 7.2 7.6 8.0

Moment magnitude, M w

Cum

ulat

ive

num

ber,

n(>

M)

per

yr

log n (>M w ) = 3.76 - 0.73M w

20 th century Radu's catalogue

M w, 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

ww

MM

w



In Eq.(5.18), the threshold lower magnitude is Mw0=6.3, the maximum credible magnitude ofthe source is Mw,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 sourcemagnitudes in the range of large recurrence intervals, where classical relationship (5.16) doesnot apply, Table 5.3.

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

Date Gutenberg-Richter

Momentmagnitude,

MRI from Eq.(5.18),

MRI from Eq.(5.16),

magnitude, MGR Mw years years8.18.07.9

-778356

150127107

10 Nov. 1940 7.47.87.77.6

217148108

917665

4 March 1977

30 Aug. 1986

7.2

7.0

7.57.47.37.2

82635040

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)

where P 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 + c1 Mw + c2 lnR +c3R +c4 h + (5.21)

where: PGA is 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 - randomvariable 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 1990’s earthquakes.

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

475

100

50



Attenuation relation - March 4, 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

PG

A, c

m/s

2 ..

median median+stdev

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

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

0

50

100

150

200

250

0 50 100 150 200 250

D, km

PG

A, c

m/s

2..

median

median+stdev

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

Attenuation relation - May 30, 1990;Mw=7.0, h=91km

0

50

100

150

200

250

300

350

400

0 50 100 150 200 250

D, km

PG

A, c

m/s

2..

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 of PGA* ;- Mmin is the mean annual rate of earthquakes of magnitude M larger or equal than Mmin;- 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 40x80km2 having the long axis oriented N45E andbeing centered at about 45.6o Lat.N and 26.6o Long. 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 k

go 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).

PSHA - Bucharest

1.E-04

1.E-03

1.E-02

1.E-01

100 150 200 250 300 350 400PGA , cm/s2

Ann

ual e

xcee

danc

e ra

te, l

(PG

A)

Figure 5.9. Hazard curve for Bucharest from Vrancea seismic source



6. ACTIONS ON STRUCTURES. SNOW LOADS

6.1 Introduction

In Europe, the norm EN 1991-1-3 gives guidance to determine the values of loads due tosnow to be used for the structural design of buildings and civil engineering works. The normdoes not apply for sites at altitudes above 1 500 m. In Romania the norm is implemented asthe law CR 1-1-3/2012 “Cod de proiectare. Evaluarea acţiunii zăpezii asupraconstrucţiilor” approved by the Romanian Ministry of Public Works and published by theOfficial Gazette of Romania.

Some aspects are not covered by the norm, for example:- impact snow loads resulting from snow sliding off or falling from a higher roof;- the additional wind loads which could result from changes in shape or size of the

construction works due to the presence of snow or the accretion of ice;- loads in areas where snow is present all year round;- ice loading;- lateral loading due to snow (e.g. lateral loads exerted by drifts);- snow loads on bridges.

Snow loads shall be classified as variable, fixed, and static actions.Depending on geographical locations for the particular condition, exceptional snow loads andthe loads due to exceptional snow drifts may be treated as accidental actions.

The relevant snow loads shall be determined for each design situation identified, inaccordance with EN 1990:2002 in Europe and CR0-2013 in Romania.

The design situations have to be considered for the two types of local site conditions:a) Normal conditions (without considering exceptional snow drifts on the roof)

The transient/persistent design situation should be used for both the undrifted and the driftedsnow load arrangements.

b) Exceptional conditions (locations considering exceptional snow drifts on theroof)

i) the transient/persistent design situation should be used for both the undrifted and the driftedsnow load;ii) the accidental design situation (the snow load is accidental action) should be used when isconsidered accidental drifted snow load.

For local effects (ex. drifting at projections and obstructions, the edge of the roof, snowfences) the persistent/transient design situation should be used.

Due to the temperate climate, for the design of buildings in Romania, the exceptionalconditions relating to areas with exceptional snow falls on the ground (characterized by a verylow probability of occurrence) are not taken into account.

6.2 Snow load on the ground

According to international practice the snow load on the ground can be studied by adopting areference period of one year to collect the statistical data, because it believes that the annual



meteorological data are statistically independent. The statistical analysis use annual maximumvalues, which in this case is the maximum associated with winter snow. Although in somegeographic regions over a long period of time can identify certain trends in climate evolution,the actual practice setting snow load on the ground they do not take into account.

Romanian law CR 1-1-3/2012, is following the SR EN 1991-1-3, and establish thatcharacteristic value of snow load on the ground is defined as the snow load on the groundbased on an annual probability of exceedence of 0,02, i.e. IMR=50 years, which is an upperfractile value of a random variable whose values are measured as annual maximum, using theGumbel probability distribution for maxima.

The Gumbel probability distribution for maxima was the EN 1991-1-3 recommendationbecause after analyzing the data available at European level when drawing up standard (2003),this probability distribution proved to be most suitable modeling snow in most Europeancountries (Switerland, Italy, Grece, Norway, Sueden, Finland, Island, Germany, France, Greatbritain) weather stations.

The characteristic values of snow load on the ground in Romania, sk, are indicated in thezoning map in Figure 6.1. The values shown are valid for designing the action for altitudes ofthe sites A ≤ 1000 m. The values in Figure 6.1 are minimum values used in the design actionof snow load.

Figure 6.1 Characteristic value of snow load on the ground zoning map, sk, kN/m2, foraltitudide of the sites A ≤ 1000 m

The altitude of the site is height above mean sea level, of the site where the structure is to belocated, or is already located for an existing structure.

For the altitude of the site 1000m < A ≤ 1500m, the characteristic value of snow load on theground is:

sk(1000m < A ≤ 1500m) = 2,0 + 0,00691 (A-1000) for sk(A≤1000m)=2,0 kN/m2 (6.1)



sk(1000m < A ≤ 1500m) = 1,5 + 0,00752 (A-1000) for sk(A≤1000m)=1,5 kN/m2 (6.2)

Ground level snow loads for any mean recurrence interval different to that for thecharacteristic snow load, sk, (which by definition is based on annual probability of exceedenceof 0,02) may be adjusted to correspond to characteristic values by application of relation 6.3.

k1

1

p sV2,5931

Vp

0,451

s

282,1)lnln(

(6.3)where:sk is the characteristic snow load on the ground (kN/m2), with a return period of 50 years-annual probability of exceedence of 0,02 (annual probability of nonexceedence p=0.98) - inaccordance with EN 1990:2002sp is the snow load on the ground having the p annual probability of nonexceedenceV1 is the coefficient of variation of annual maximum snow load (in Romania V1 is 0,35÷1,0).

Figure 6.2 are exemplified ratios of the snow load on the ground having MRI = 75years,respectively MRI = 100years and with characteristic snow load on the ground (MRI=50years) for different values of the coefficient of variation V1.

1.05

1.1

1.15

1.2

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Coeficientul de variatie V1

Rap

ortu

l sIM

R/s I

MR=

50an

i

sIMR=75ani/sIMR=50ani

sIMR=100ani/sIMR=50ani

Figure 6.2 Ratios of snow load on the ground having MRI = 75years and MRI = 100 yearsand with characteristic snow load on the ground (MRI=50 years)

The exceptional snow load on the ground is the load of the snow layer on the groundresulting from a snow fall which has an exceptionally infrequent likelihood of occurring – dueto the temperate climate is not applicable for Romania. If is the case, the EN 1991-1-3recommended to double value of the sk.

The bulk weight density of snow varies. In general it increases with the duration of the snowcover and depends on the site location, climate and altitude. Indicative values for the meanbulk weight density of snow on the ground given in Table 6.1 may be used.



Table 6.1: Mean bulk weight density of snowType of snow Bulk weight density [kN/m3]Fresh 1,0Settled (several hours or days after its fall) 2,0Old (several weeks or months after its fall) 2,5 - 3,5Wet 4,0

6.3 Snow load on the roof

There is still relatively little data from measurements on the snow load on the roof andmeasurement procedures are not standardized. In addition, there are practical difficulties ofmaking multiple measurements. Therefore uncertainties associated with the snow load on theroof are greater than the uncertainties associated with the snow load on the ground. It hashighlighted the existence of a large number of types, different roofs. The rules and codes triesto collect and standardize the types of roofs, but obviously it is impossible to be considered allpossible configurations for them. The design shall recognise that snow can be deposited on aroof in many different patterns. Properties of a roof or other factors causing different patternscan include:

a) the shape of the roof;b) its thermal properties;c) the roughness of its surface;d) the amount of heat generated under the roof;e) the proximity of nearby buildings;f) the surrounding terrain;g) the local meteorological climate, in particular its windiness, temperature, variations,

and likelihood of precipitation (either as rain or as snow).

In regions with possible rainfalls on the snow and consecutive melting and freezing, snowloads on roofs should be increased, especially in cases where snow and ice can block thedrainage system of the roof.The load should be assumed to act vertically and refer to a horizontal projection of the roofarea.When artificial removal or redistribution of snow on a roof is anticipated the roof should bedesigned for suitable load arrangements.

The following two primary load arrangements shall be taken into account:1. undrifted snow load on the roof - load arrangement which describes the uniformly

distributed snow load on the roof, affected only by the shape of the roof, before anyredistribution of snow due to other climatic actions).

2. drifted snow load on the roof - load arrangement which describes the snow loaddistribution resulting from snow having been moved from one location to anotherlocation on a roof, e.g. by the action of the wind).

The characteristic value of snow load on the roof is the product of the characteristic snowload on the ground and appropriate coefficients.

The snow loads on roofs for the persistent/transient design situations shall be determined asfollows:

s = Is i Ce Ct sk (6.2)



where:Isis the importance/exposure coefficient for the snow load;μi is the snow load shape coefficient;sk is the characteristic value of snow load on the groundCe is the exposure coefficientC t is the thermal coefficient.

The snow loads on roofs for the accidental design situations where exceptional snow drift isthe accidental action shall be determined as follows:

s = Is i sk (6.3)where:

Isis the importance/exposure coefficient for the snow load;μi is the snow load shape coefficient;sk is the characteristic value of snow load on the ground

The exposure coefficient Ce should be used for determining the snow load on the roof. Thechoice for Ce should consider the future development around the site. Ce should be taken as1,0 unless otherwise specified for different topographies. He characterizes the overall effect ofthe wind on snow deposit on the building by the surrounding topography and naturalenvironment and / or built in the vicinity of the building.

Table 6.2 Values of Ce for different topographies

Topographies Ce

Windsweptflat unobstructed areasexposed on all sideswithout, or little shelterafforded by terrain, higherconstruction works or trees

0,8

Normalareas where there is nosignificant removal ofsnow by wind onconstruction work,because of terrain, otherconstruction works ortrees.

1,0

Shelteredareas in which theconstruction work beingconsidered is considerablylower than the surroundingterrain or surrounded byhigh trees and/orsurrounded by higherconstruction works

1,2

The thermal coefficient C t, is the coefficient defining the reduction of snow load on roofs asa function of the heat flux through the roof, causing snow melting. The thermal coefficient C t



should be used to account for the reduction of snow loads on roofs with high thermaltransmittance (> 1 W/m2K) for some glass covered roofs, because of melting caused by heatloss. Based on the thermal insulating properties of the material and the shape of theconstruction work the thermal coefficient value is determined by special studies, see ISO4355 - 1998. Ct should be taken as 1,0.

The roughness of the roof surface influences the snow sliding and is difficult to be evaluated.For example, in some areas of the roof, the small objects might prevent natural snow slide(other than parapets). However, since it is considered that the snow slides off the roof entirely(when there are no obstacles or parapets) if the roof angle above 60°, those shape coefficientsare zero for these parts of roofs.

6.4 Roof shape coefficients

The roof snow load shape coefficient is ratio of the snow load on the roof to the undriftedsnow load on the ground, without the influence of exposure and thermal effects.

European values of coefficients of EN 1991-1-3 were calibrated based on the analysis resultsof experimental studies both on site (in-situ) and in the wind tunnel and the comparativeanalysis of coefficients prescriptions from different countries, see Formichi, P., 2008. “EN1991 – Eurocode 1: Actions on structures. Part 1-3 General actions – Snow Loads”,presentation at Workshop “Eurocodes. Background and Applications”, 18-20 Feb., Brussels,60p.

The results of in-situ measurements in the US, Canada, Norway and England werecompleted with the results of measurements conducted special campaigns in Europe forstudying deposits of snow on roofs (winter 1998/1999). The measurements were verydetailed, both in terms of meteorological parameters (wind speed, wind direction, airtemperature, air humidity, solar radiation, rainfall regime, etc.) and in terms of roof types(shape dimensions, inclinations, the surface heat transfer from inside the building, roofinsulation, etc.), altitude, exposure (wind, sun), drifts of snow on the roof at various points,etc. In England measurements were performed on 25 types of roofs in 18 different sites ataltitudes from 5m to 656 m. In the Italian Alps measurements were performed on 13 rooftopsin 7 different sites at altitudes from 88m to 1340m and in the Dolomites on sites. In Germanymeasurements were performed on three roofs in 2 different sites at altitudes of 141m and880m, and in Switzerland on 35 rooftops at 8 different sites at altitudes from 570m to 1628m.In total, 81 measurements were carried out on the roof, see Sanpaolesi L., 1999. ScientificSupport Activity In The Field Of Structural Stability Of Civil Engineering Works - SnowLoads, Final Report, Commission Of The European Communities, DGIII - D3, Contract n°500990/1997, 172p. These in-situ measurement information were completed with results oflaboratory tests at "Jules Verne climatic tunnel" Centre Scientifique et Technique du'sBâtiment (CSTB), Nantes.

The snow load shape coefficients that should be used for roofs are given in Table 6.3 and inFigure 6.3. Where snow fences or other obstructions exist or where the lower edge of the roofis terminated with a parapet, then the snow load shape coefficient should not be reducedbelow 0,8.



Table 6.3 Snow load shape coefficients used for monopitch roofs, pitched roofs and for multi-span roofs

Angle of pitch of roof, 0 00 300 300 < < 600 600

1 0,8 0,8 (60 - )/30 0,0

2 0,8 + 0,8 /30 1,6 -

Figure 6.3 Snow load shape coefficients used for monopitch roofs, pitched roofs and formulti-span roofs

0.51(2)

1(2)0.51(1)

1(1)

1(2)1(1)

Figure 6.4 Snow load shape coefficients and the load arrangements used for monopitch roofs,pitched roofs and for multi-span roofs

In the case of monopitch roofs, the load arrangement of Figure 6.4 should be used for both theundrifted and drifted load arrangements.

In the case of pitched roofs the load arrangement of Figure 6.4 are as follows:- The undrifted load arrangement which should be used is case (i);- The drifted load arrangements which should be used are cases (ii) and (iii), unless

specified for local conditions.

undrifted snow on pitched roofs drifted snow on pitched roofs

Case (i)Case (ii)Case (iii)



For multi-span roofs the load arrangement of Figure 6.4 are as follows:- The undrifted load arrangement which should be used is shown in case (i);- The drifted load arrangement which should be used is shown in case (ii), unless


undrifted snow on multi-span roofs drifted snow on multi-span roofs

The snow load shape coefficients that should be used for cylindrical roofs, in absence of snowfences, are given in the Figure 6.5.

3

ls

0,53

0,8

ls/4 ls/4 ls/4 ls/4

h

b

Cazul (i)

Cazul (ii)

Figure 6.5 Snow load shape coefficients for cylindrical roof

For cilindrical roofs the load arrangement of Figure 6.5 are as follows:- The undrifted load arrangement which should be used is shown in case (i);- The drifted load arrangement which should be used is shown in case (ii), unless


6.5 Local effects

6.5.1 Local verification

For the local verifications the applied forces used in the design situations have to beconsidered are persistent/transient.

In windy conditions drifting of snow can occur on any roof which has obstructions as thesecause areas of aerodynamic shade in which snow accumulates.



The snow load shape coefficients and drift lengths for quasi-horizontal roofs should be takenas follows (see Figure 6.1), unless specified for local conditions:

1 = 0,82 = h / sk with restrictions 0,8 2 2,0 - weight density of snow (2 kN/m3)ls = 2 h with restrictions 5 m ls 15 m.

The design of those parts of a roof cantilevered out beyond the walls should take account ofsnow overhanging the edge of the roof, in addition to the load on that part of the roof. Theloads due to the overhang may be assumed to act at the edge of the roof and may be calculatedas follows:

se = k s2/ se - snow load per metre length due to the overhangs - the most onerous undrifted load case - weight density of snow (3 kN/m3)k - a coefficient to take account of the irregular shape of thesnowk = 3/d with restrictions k ≤ d ,

6.5.2 Exceptional snow drift on the roof

The load due to exceptional snow drift is the load arrangement which describes the load of thesnow layer on the roof resulting from a snow deposition pattern which has an exceptionallyinfrequent likelihood of occurring.

It should be assumed that they are exceptional snow drift loads and that there is no snowelsewhere on the roof.

The accidental design situation (the snow load is accidental action) should be used when theload is considered as accidental drifted snow load.

The snow load shape coefficients used for evaluation of exceptional snow drift on the roof aregiven in the Annex B of the EN 1991-1-3:2003 or in the Chapter 7 of the CR 1-1-3/2012.

d

se



7. ACTIONS ON STRUCTURES. WIND ACTION

The wind action on buildings and structures is presented according to the EUROCODE 1:Actions on structures — Part 1-4: General actions — Wind actions and in the Romaniandesign code CR 1-1-4/2012.

7.1. General

Wind effects on buildings and structures depend on the exposure of buildings, structures andtheir elements to the natural wind, the dynamic properties, the shape and dimensions of thebuilding (structure).

Wind actions fluctuate with time and act directly as pressures on the external surfaces ofenclosed structures and, because of porosity of the external surface, also act indirectly on theinternal surfaces. They may also act directly on the internal surface of open structures.Pressures act on areas of the surface resulting in forces normal to the surface of the structureor of individual cladding components. Additionally, when large areas of structures are sweptby the wind, friction forces acting tangentially to the surface may be significant.

The wind action is represented by a simplified set of pressures or forces whose effects areequivalent to the extreme effects of the turbulent wind.

The wind actions calculated using EN 1991-1-4 are characteristic values. They are determinedfrom the basic values of wind velocity or the velocity pressure having annual probabilities ofexceedence of 0,02, which is equivalent to a mean return period of 50 years.

The wind velocity and the velocity pressure are composed of a mean and a fluctuatingcomponent. The random field of natural wind velocity is decomposed into a mean wind in thedirection of air flow (x-direction) averaged over a specified time interval and a fluctuatingand turbulent part with zero mean and components in the longitudinal (x-) direction, thetransversal (y-) direction and the vertical (z-) direction.

0

0

V(z,t) = vm(z) + v(z,t)

Mean wind velocity

vm(z)

Averaging time intervalof mean velocity (10 min)

t

v(z,t)V(z,t)

vm(z)

t

v(z,t)

Gusts, velocity fluctuations from the mean



7.2 Reference wind velocity and reference velocity pressure

The fundamental value of the basic wind velocity, vb , is the characteristic 10 minutes meanwind velocity, irrespective of wind direction and time of year, at 10 m above ground level inopen and horizontal terrain exposure (category II) with an annual risk of being exceeded of0, 02 (the mean recurrence interval MRI=50years).

For other than 10 min averaging intervals, in open terrain exposure, the followingrelationships may be used:

3sb

1minb

10minb

1hb 67,00,8405,1 vvvv (7.1)

The coefficient of variation of maximum annual wind velocity, V1 = 1 / m1 depends on theclimate and is normally between 0.10 and 0.40; the mean of the maximum annual windvelocities is usually between 10 m/s to 50 m/s. The sequence of maximum annual mean windvelocities can be assumed to be a Gumbel distributed sequence with possibly directiondependent parameters.

The reference wind velocity pressure can be determined from the reference wind velocity(standard air density =1.25kg/m3) as follows:

smvvPaq bbb /625.02

1 22 (7.2)

The conversion of velocity pressure averaged on 10 min into velocity pressure averaged onother time interval can be computed from relation (9.1):

sbqqq 31min

b10minb

1hb 44.00.7q1.1 . (7.3)

The reference wind velocity pressure qb (in kPa) with mean recurrence interval MRI=50 yr.averaged on 10 min. at 10 m above ground in open terrain is presented in Figure 7.1.

Figure 7.1. The reference wind velocity pressure, [kPa] with MRI=50 yr. –wind velocity averaged on 10 min. at 10 m above ground in open terrain. The mapped values

are valid for sites situated bellow 1000 m



The mean wind velocity vm(z) at a height z above the terrain depends on the terrain roughnessand orography and on the basic wind velocity, vb and should be determined using Expression(7.4).

vm(z) = co(z) )(zcr vb (7.4)where:

co(z) - is the orography factor, taken as 1,0)(zcr -is the roughness factor

7.3 Terrain roughness and Variation of the mean wind with height

The roughness of the ground surface is aerodynamically described by the roughness length, zo,(in meters), which is a measure of the size of the eddies close to the ground level. Variousterrain categories are classified in Table 7.1 according to their approximate roughness lengths.

Table 7.1. Roughness length zo, in meters, for various terrain categories 1) 2)

Terraincategory

Terrain description zo [m] zmin [m]

0 Sea or coastal area exposed to the open sea 0.003 1I Lakes or flat and horizontal area with negligible vegetation and

without obstacles0.01 1

II Area with low vegetation such as grass and isolated obstacles(trees, buildings) with separations of at least 20 obstacle heights

0.05 2

III Area with regular cover of vegetation or buildings or withisolated obstacles with separations of maximum 20 obstacleheights (such as villages, suburban terrain, permanent forest)

0.3 5

IV Area in which at least 15 % of the surface is covered withbuildings and their average height exceeds 15 m

1.0 10

1) Smaller values of zo produce higher mean velocities of the wind

2) For the full development of the roughness category, the terrains of types 0 to III must prevail in theup wind direction for a distance of at least of 500m to 1000m, respectively. For category IV thisdistance is more than 1 km.

minmin

maxmin0

f

m200zfln

zzorzc

zzorz

zk

zc

r

r

r (7.5)

Where,

kr is terrain factor depending on the roughness length calculated with:

07,0

0

05,0189,0

zkr (7.6).

z0 and zmin values are given in the Table 7.1. The values kr are indicated in Table 7.2.



Table 7.2. The kr şi kr2 factors for various terrain categories

Terraincategory

0 I II III IV

kr 0,155 0,169 0,189 0,214 0,233kr

2 0,024 0,028 0,036 0,046 0,054

(a) (a)

(b) (b)

Figure 7.2a Category 0 - Sea or coastal area exposedto the open sea (z0 = 0,003 m)

Figure 7.2b Category II - Area with low vegetationsuch as grass and isolated obstacles (trees,

buildings) with separations of at least 20 obstacleheights (z0 = 0,05 m)

(a) (a)

(b) (b)Figure 7.2c Category III - Area with regular cover

of vegetation or buildings or with isolated obstacleswith separations of maximum 20 obstacle heights

(such as villages, suburban terrain, permanentforest) (z0=0,3 m)

Figure 7.2d Categoria IV - Area in which at least15 % of the surface is covered with buildings and

their average height exceeds 15 m (z0 = 1,0 m)



The logarithmic profile of the mean wind velocity is valid for moderate and strong winds(mean velocity > 10 m/s) in neutral atmosphere (where the vertical thermal convection of theair may be neglected).

The roughness factor is represented in Figure 7.2.

0

20

40

60

80

100

120

140

160

180

200

0 0.5 1 1.5 2 2.5 3 3.5

Hei

ght a

bove

gro

und

z, m

Roughness factor, cr(z)

Terrain category 0

Terrain category I

Terrain category II

Terrain category III

Terrain category IV

Figure 7.2. Roughness factor,

00 z

zln)z(k)z(c rr

Consequently, the mean wind velocity pressure at height z is defined by:

b22

m qzcczq ro (7.7)

Where:

minmin2

maxmin

2

00

2

2

f

m200zln

zzorzzc

zzforz

zzk

zc

r

r

r (7.8)

zref=10m

z Wind velocity will be the same(almost constant)

Atmospheric boundary layerThe mean wind velocity is increasing - gradient height (2km)

(almost constant)



7.4 Wind turbulenceThe turbulence intensity Iv is measure of the fluctuations of wind velocity around the meanvalue. The turbulence intensity Iv (z) at height z is defined as the standard deviation of theturbulence (of the wind velocity v(z,t)) divided by the mean wind velocity at height z, (vm(z)).

zvzI

m

vv

(7.9)

The recommended rules for the determination of Iv (z) are given in Expression (7.10):

minminv

maxmin

0v

f

m200z

ln5,2

zzorzzI

zzfor

z

zzI

(7.10)

The values of (Table 7.3) depends of terrain roughness factor (z0, m):

5,7ln856,05,45,4 0 z (7.11)

Table 7.3. Values of

Terrain category 0 I II III IV

2,74 2,74 2,66 2,35 2,12

7.5 Peak values

The peak value of the wind velocity is the maximum expected value to occur within10minutes of averaging period. Considering a normal distribution of the wind velocity, thepeak velocity is:

zIgzvgzvzv vmvmp 1 (7.12)

zvzczv mpvp ; (7.13)

zIzIgzv

zvzc vv

m

ppv 5,311 (7.14)

Where:

g is the peak factor, recommended value g=3,5

cpv(z) is gust factor for mean wind velocity.

The peak velocity pressure qp(z) at height z, which includes mean and short-term velocityfluctuations, should be determined by:

zIzqzIgzvzq vmvmp 7112

1 22



The gust factor for velocity pressure is the ratio of the peak velocity pressure to the meanwind velocity pressure.

zIzIgzq

zqzc vv

m

ppq 7121 (7.15)

The peak velocity pressure at the height z above ground is the product of the gust factor, theroughness factor and the reference velocity pressure.

b2

pqmpqp qzczczqzczq r (7.16)

bp qzczq e (7.17)

The exposure factor is defined as the product of the gust and roughness factors:

zczczc re2

pq (7.18)

If we cannot neglect the orography effect, the exposure factor ce(z) is:

zczcczc re pq22

0 (7.19)

0

20

40

60

80

100

120

140

160

180

200

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Inal

tim

ea d

easu

pra

tere

nulu

iz,

m

Factorul de expunere, ce(z)

Teren categoria 0

Teren categoria I

Teren categoria II

Teren categoria III

Teren categoria IV

Figure 7.3 Illustrations of the exposure factor ce(z)7.6 Wind actions

Wind action is the product of wind pressures on the surfaces of buildings and structures, orthe forces produced by wind on buildings and structures. The wind actions are variableactions in time and act both directly as pressure / suction on exterior surfaces of buildings andenclosed structures and indirectly on the inner surfaces of buildings and other facilities closeddue to porosity outer surfaces. The pressure / suction and may act directly on the innersurfaces of buildings and other open facilities. The pressure / suction forces are acting normalon the surface of construction. In addition, when large areas of the buildings are exposed to



the wind, the horizontal frictional forces acting tangentially to the surface can have asignificant effect.

The wind action is classified as a fixed variable action; actions are assessed as wind pressure /suction or forces and are represented by their characteristic values.

Wind actions on structures and structural elements shall be determined taking account of bothexternal and internal wind pressures.

7.6.1 Wind pressure on surfaces

The wind pressure acting on the external surfaces, ee zqcw ppeIw (7.20)

where:qp(ze) is the peak velocity pressure;ze is the reference height for the external;cpe is the pressure coefficient for the external;Iw importance– exposure factor for wind actions.

The wind pressure acting on the internal surfaces of a structure, wi, should be obtained from ii zqcw ppiIw (7.21)

where:qp(zi) is the peak velocity pressure ;zi is the reference height for the internal pressure;cpi is the pressure coefficient for the internal pressure.

Figure 7.4 Pressure on surfaces

The net pressure on a wall, roof or element is the difference between the pressures on theopposite surfaces taking due account of their signs. Pressure, directed towards the surface istaken as positive, and suction, directed away from the surface as negative. Examples are givenin Figure 7.4.



7.6.2 Wind forces

The wind forces for the whole structure or a structural component should be determined:- by calculating forces using force coefficients (i)or- by calculating forces from surface pressures (ii).

(i) The wind force Fw acting on a structure or a structural component may be determineddirectly by using Expression 7.22:

refpfdIww AzqccF e (7.22)

or by vectorial summation over the individual structural elements by using Expression (7.23):

elements

refpfdIww AzqccF e (7.23)

Where:qp(ze) is the peak velocity pressure reference height ze;cd is the structural factor;cf is the force coefficient for the structure or structural element;Aref is the reference area of the structure or structural element.

(ii) The wind force, Fw acting on a structure or a structural element may be determined byvectorial summation of the forces Fw,e and Fw,i, calculated from the external and internalpressures using Expressions (7.24) and (7.25) and the frictional forces resulting from thefriction of the wind parallel to the external surfaces, calculated using Expression (7.26).

external forces:

surfaces

refeedew AzwcF , (7.24)

internal forces:

surfaces

refiiiw AzwF , (7.25)

friction forces:

frefrIwfr AzqcF p (7.26)

where:

cd is the structural factor;we(ze) is the external pressure on the individual surface at height ze;wi(zi) is the internal pressure on the individual surface at height zi;Aref is the reference area of the individual surface;cfr is the friction coefficient;Afr is the area of external surface parallel to the wind.



8. ACTIONS ON STRUCTURES. SEISMIC ACTION

8.1 IntroductionFor implementing the EN 1998, national territories are subdivided into seismic zones,depending on the local hazard. By definition, the hazard within each zone is assumed to beconstant. For most of the applications of EN 1998, the hazard is described in terms of a singleparameter, i.e. the value of the reference peak ground acceleration.

Seismic hazard level indicated by the National Authorities is a minimum for design.

In Romania, the reference peak ground acceleration, chosen by the National Authorities foreach seismic zone, corresponds to the reference return period T=225years, requirement (orequivalently the reference probability of exceedance in 50 years, P=20%).

Figure 8.1 Zonation of design peak ground acceleration, ag

The AEd, design value of seismic action is equal with AEk, characteristic value of the seismicaction for the reference return period multiplied with the importance/exposure factor of theconstruction:

dEA = γI,e ·kEA (8.1)

8.2 Representation of the seismic action

Within the EN 1998, the earthquake motion at a given point on the surface is represented byan elastic ground acceleration response spectrum, henceforth called an “elastic responsespectrum. The horizontal seismic action is described by two orthogonal components assumedas being independent and represented by the same response spectrum.The elastic response spectrum Se (T) is defined by the following expressions:



Ta)T(S ge (8.2)

whereSe (T) is the elastic response spectrum;T is the vibration period of a linear single-degree-of-freedom system;ag is the design ground acceleration (m/s2), from zonation map; T is the normalized elastic response spectrum of absolute accelerations.

In the case of 0,05% viscous damping, the normalized elastic response spectrum of absoluteaccelerations is defined by the following relations, Figure 8.3:

0 T TB

TT

11(T)

B

0

(8.3)

TB < T TC 0 (8.4)

TC < T TDT

T(T) C

0 (8.5)

TD < T 5s20

T

TT(T) DC (8.6)

where:0 maximum dynamic amplification factor of the ag by the structure, 0 =2,5;T is the vibration period of a linear single-degree-of-freedom system;TB is the lower limit of the period of the constant spectral acceleration branch;TC is the upper limit of the period of the constant spectral acceleration branch;TD is the value defining the beginning of the constant displacement response range

of the spectrum.

The values of the periods TB, TC , TD describing the shape of the elastic response spectrumdepend upon the ground type. In Romania, based on seismic records, the zonation of controlperiod TC (s) is given in Figure 8.2.

Figure 8.2 Zonation of control period TC (s)



Table 8.1 Values of the periods TB, TC , TD describing the horizontal elastic response spectra

TC (s) 0,7 1,0 1,6TB (s) 0,14 0,2 0,32TD (s) 3,0 3,0 2,0

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3 3.5 4Perioada T , s

T D =3

5.25/T 2

1.75/T

= 2.5

T B =0.14

T C=0.7s

=0,05

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3 3.5 4Perioada T , s

T D =3T C=1.0s

7.5/T 2

2.5/T

=2.5

T B =0.2

=0,05

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3 3.5 4Perioada T , s

T D =2

8/T 2

4/T

=2.5

T B =0.32 T C=1.6s

=0,05

Figure 8.3 Shapes of normalized elastic response spectrum for TC = 0,7s, 1,0s and 1,6s (5%damping)

(T

)

(T)

(T

)



The elastic displacement response spectrum, SDe(T) (in meters), shall be obtained by directtransformation of the elastic acceleration response spectrum, Se(T), using the followingexpression:

2

2

T

)T(S)T(S eDe (8.7)

8.3 Design spectrum for elastic analysis

The capacity of structural systems to resist seismic actions in the non-linear range generallypermits their design for resistance to seismic forces smaller than those corresponding to alinear elastic response. To avoid explicit inelastic structural analysis in design, the capacity ofthe structure to dissipate energy, through mainly ductile behaviour of its elements and/or othermechanisms, is taken into account by performing an elastic analysis based on a responsespectrum reduced with respect to the elastic one, henceforth called a ''design spectrum''. Thisreduction is accomplished by introducing the behaviour factor q.

Behavior factor q is a factor used for design purposes to reduce the forces obtained from alinear analysis, in order to account for the non-linear response of a structure, associated withthe material, the structural system and the design procedures. The values of the behaviourfactor q, which also account for the influence of the viscous damping being different from5%, are given for various materials and structural systems according to the relevant ductilityclasses in the various Parts of EN 1998.

For the horizontal components of the seismic action the design spectrum, Sd(T) (in m/s2) shallbe defined by the following expressions:

0 < T TB

T

T

qa)T(S

Bgd

11

0

(8.9)

T > TB Sd (T) gg a,q

)T(a 20

(8.10)

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