## Approximation of the factors kn and kdn

Factors kn and kd,n can be calculated by interpolation from the values in Tables 2 and 3 or, alternatively, by approximation functions kn 1,655+0,672 n , p 0,05, Vx known kn n (-0,950+0,614xn) , p 0,05, Vx unknown kn 3,099+1,294 n , p 0,001, Vx known kn n (-0,986+0,323xn) , p 0,001, Vx unknown Figure 3 shows how these formulas approximate the data from the Tables 2 and 3. The error when using these formulas is typically less then 1 . Figure 3. Approximation functions for the factors kn and kd,n...

## Deterministic approach the global load factor

Concept of the global load factor is sometimes used to compare various alternatives for load combination with no regard to a resistance of a structure. The deterministic global load factor yE follows directly from codified combination rules and given partial factors without any probabilistic consideration it is simply expressed as where the design load effect Ed is given by one of equations (1) to (4) depending on the combination rule considered (for example combination rules A, B or C). It...

## DPf PA n B Pa Pb Px E xdx PR x Orx x dx

The above-mentioned assumption of mutual independence of the variables E and R, and thus also of the events A and B, is applied here. Figure 5. Distribution of variables E and R. Figure 5. Distribution of variables E and R. The integration of the differential relationship (20) over the interval in which both variables E and R occur simultaneously (generally the interval < -w, w> ) leads to the relation the integration of the relation (21) usually has to be carried out numerically or using...

## Population parameters

The population parameters are quantities used in describing the distribution of a random variable, as estimated from one or more samples. As in the case of random samples, three basic population parameters are commonly used in practical applications - the mean p representing the basic measure of central tendency - the variance o as the basic measure of dispersion and - the coefficient of skewness .giving the degree of asymmetry. The population mean , for a continuous variable X having the...

## Relative values of fractiles xPX versus probability P Lower fractiles

Notes. 1) It follows from Figure that the skewness of the distribution may have significant effect on assessment of the design value (0,001 fractile). 2) Approximate formula for two parameter lognormal distribution yields sufficiently accurate results for the coefficient of variation V < 0,2. 3) Gamma and Gumbel distribution can be well approximated by three parameter lognormal distribution having skewness equal to a 2* V and a 1,14 respectively. Attachment 2 - MATHCAD sheet SampFract.mcd...

## Reliability Ii

Milan Holicky , Ton Vrouwenvelder and Angel Arteaga 1Klokner Institute, Czech Technical University in Prague, Czech Republic 2Delft University of Technology, TNO BOUW, The Netherlands Institute of Construction Sciences 'E. Torroja', CSIC, Madrid, Spain Using basic principles of the reliability theory described in Chapter II Elementary methods of structural reliability I, the operational techniques for estimating partial factors of basic variables are derived and applied to common permanent and...

## XJdi

A n-c g a, b + d g a, g J (3.15) where g is an auxiliary parameter. From equations (3.15), relations for parameters c and d can be derived c S-a r (u- a)(b _ 1j d b - Q- a)(b - q) _ For the moment parameters of the beta distribution it holds that i a + (b-a)c , a (b-a) (3.17) 2g(d - c) 3g2(2(c + d)2 + cd(c + d - 6)) - 3 (318) (c + d + 2) ' (c + d + 2)(c + d + 3) ' Note that skewness a> and kurtosis s are dependent only on the parameters c and d (they are independent of the limits a and b)....

## P

Variation of pn with p1 for n 5, 25, 50 and 100 Note that, if 1-year period would be used for specification of the target reliability level of a structure, then Figure 1 provides information on the resulting failure probability corresponding to a given working life Tn. For example, if the target reliability level is specified by the reliability index p1 4,7 (corresponding to the probability p1 1,3 x 10-6), then (as already mentioned) the reliability level of a structure having a...

## Design Assisted By Testing

Institute for Metal Constructions, Ljubljana, Slovenia Under particular circumstances it may be favorable or necessary to carry out tests in order to obtain certain design parameters. Typical parameters determined from the tests are actions on the structure, resistance of the structure or structural component and material properties. Tests can be performed also to calibrate parameters in the theoretical model of resistance. The design value of the parameter is obtained from the test results as...

## Reliability differentiation Chapter Iii Reliability Differentiation

Milan Holicky1-' and Jana Markova1-1 1)Czech Technical University in Prague, Czech Republic Basic reliability elements specified in current standards for structural design commonly include failure probability related to a certain reference period T. Required reliability level of buildings and other civil engineering works is usually specified by the design (target) failure probability pd or by appropriate reliability index Jd corresponding to a specified design working life Td (for example 50...

## Structural Reliability Info

Coefficients kp and -tp(1 n + 1) forp 0,05 and normal distribution of the Also the skewness (asymmetry) of the population o may affect significantly the estimator of the population's fractile. Tables 4.5 and 4.6 show the coefficients kp from equation (4.8) for three value of the skewness o -1,0, 0,0 and 1,0, probability p 0,05 and confidence y 0,75 (Table 4.5) and y 0,95 (Table 4.6). Values of the coefficients from Table 4.6 are shown in Figure 4.4. Table 4.5. Coefficient kp from...

## Reliability I

1 2 Milan Holicky and Ton Vrouwenvelder 1Klokner Institute, Czech Technical University in Prague, Czech Republic 2Delft University of Technology, TNO BOUW, The Netherlands Elementary methods of structural reliability are described considering a fundamental case of two random variables when the limit state function is formulated as a difference between the resulting structural resistance and load effect. The initial assumption of normal distribution of both resulting variables is generalised to...

## One variable action

Results of the reliability analyses are presented in graphical form that indicates variation of the reliability index failure probability Pf, and sensitivity factors aR, aE, aG, aQ and aW with the load ratio x In particular Figure 3 shows results of a simple case of one variable action only the main variable action Q Figure 3 indicates the variation of - for expression 6.10 of EN 1990 sensitivity factors aR, aE, and partial sensitivity factors aG, aQ and aW For the analysis it has been assumed...

## Probability density u

The lower and upper fractiles of a standardized random variable U having normal Figure 4.2. The lower and upper fractiles of a standardized random variable U having normal In the case of a lognormal distribution with lower limit at zero, which is described in section 3.2, it is possible to calculate the fractile from the value of fractile of a standardized random variable with normal distribution using the relation 7 TeXp u 0rm,,Vln 1 V2 4.3 where wnorm,p is the fractile of a...

## Generic Structural Member

In case of generic structural member it is assumed that the characteristic value Rk of the resistance R may be defined as the 5 fractile of R and the design value Rd as where yR denotes the global resistance factor commonly expected to be within the range from 1 to 1,2 . The significance of both values Rk and Rd is obvious from Figure 2, where the random variable R is described by the probability density function R , and the design value Rd is indicated as a particular value of R corresponding...

## Info

Event constitutes the starting point. Going out from this event, possible causes are to be identified. The possible causes and consequences are to be linked in a logic way, without introducing any loops. Every event that is not a consequence of the previous event has to be considered as an independent variable. An example of the fault tree shown in Figure 3 describes the failure of a plane frame indicated at the bottom of Figure 3 . Figure 3. Fault tree describing the failure of a plane frame....

## Mathcad sheet Beta Time

Mathcad sheet Beta-Time is intended for transformation of probability and reliability index Beta for different reference periods n 1,n -qnorm _1 - l - pnorm - 1,0, l n 1, 1 n 1,5 n 1, 50 nt 2 nt 3.8 pnt p1 nt pnt 7.235x 10 nt 3.8 pnt p1 nt pnt 7.235x 10 n 1, 1 n 1,5 n 1, 50 nt 2

## Appendix C Notation

Load effect including model uncertainty load effect without model uncertainty characteristic value of the load effect E permanent load including model uncertainty, G 0 G0 permanent load without load uncertainty design value of the resistance G, Gd YG Gk characteristic value of the permanent load G main dominant variable load including model uncertainty, Q 0Q0 main dominant variable load without model uncertainty design value of the variable load Q, Qd YQ Qk characteristic value of the variable...

## Reinforced Concrete Beam Or Slab

Partil factor or global safety factor method Bending moment qk kN m 3.00 gammaQ 1.5 Concrete fck MPa 20 Rebars fyk MPa 500 x d lt max 0.45 Estimate z 0,9 d P P gt pmin x d lt max General Table acc 1 icc fck yc 13.3 yd fyk ya 434.8 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 Attachment 6 - MATHEMATICA notebook Fitdistefouton nb

## Combination factor w for accompanying action

1 Input data V is coefficient of variability of the accompanying action related to the reference period T 50 years , r T T1 where T1 is the greater of the of basic periods actions to be combined for example 5, 7, 10, 50 Range variables v 0.0, 0.05 1.0 r 1 50 p 3.8 reliability index 2 Factor y0 for normal distribution Formula following Turkstra's rule y0 F-1 o 0,4 0,7 p r F-1 0,7 p 1 qnorm pnorm 0.28-p, 0,1 , 0, 1 -V y0n V, r Approximation in EC 1990 3 Factor y0 for Gumbel distribution 1 -...

## Appendix A The derivation of the equation

Then the distribution ln X is normal with mean lnX lnX and standard deviation alnX. The characteristic value of ln X can be written Since the mean u X of X can be expressed with the mean lnX and standard deviation alnXof ln X by the relationship Xk X exp - knOlnX - OlnX2 2 A.4 If X Y Z is a product of two influences, Y and Z, then The standard deviation olnX of ln X can be expressed using FORM factors aY and aZ as If we now combine equations A.7 and A.4 , then...

## Statistical Determination Of Resistance Models

The procedures given in this section are intended for the calibration of resistance models and for the derivation of design values from the tests undertaken to reduce uncertainties in parameters of the resistance model. Based on observations and theoretical considerations, a design model of the resistance is developed. The statistical interpretation of the test results should then be used to validate and adjust the model, until sufficient correlation between test and theoretical data is...

## Handbook Reliability Backgrounds

The Klokner Institute of the Czech Technical University in Prague KI CTU , convener, Prof. Milan Holicky, Czech Republic The Czech Chamber of Certified Engineers and Technicians Engaged in Construction CKAIT , Prof. Alois Materna, Czech Republic, The Institute of Steel Constructions of the University of Technology Aachen RWTH , Prof. Gerhard Sedlacek, Germany The Spanish Organisation for Scientific Research IET , Spain, Dr. Angel Arteaga The University of Pisa UOP , Prof. Luca Sanpaolesi, Italy...

## Basic Concepts Of Structural Reliability

1 2 Milan Holicky and Ton Vrouwenvelder 1Klokner Institute, Czech Technical University in Prague, Czech Republic 2Delft University of Technology, TNO BOUW, The Netherlands Uncertainties affecting structural performance can never be entirely eliminated and must be taken into account when designing any construction work. Various design methods and operational techniques for verification of structural reliability have been developed and worldwide accepted in the past. The most advanced operational...

## MATLAB package Levelm

MATLAB package Level2.m is intended for determining the reliability index using FORM method. 9. MATHCAD sheet FORM2.mcd MATHCAD sheet FORM2.mcd is intended for calculation of the reliability index P and failure probability assuming function g X R - E 0 assuming general three parameter lognormal distribution LN ,a,a of E and R. 10. MATDCAD sheet FORM7.mcd MATHCAD package FORM7.mcd is intended for calculation of the reliability index P and failure probability assuming a non-linear limit state...

## Skewness Distribution

The probability density function of a normal and lognormal distribution with a coefficient of skewness c 1,0 described in the next section 3.2 of the standardized random variable u is shown in Figure 3.1. Note that the probability density function of the standardized normal distribution is plotted in Figure 3.1 for u in the interval lt -3, 3 gt , which covers the standardised variable U with a high probability of 0,9973 in engineering practice this interval is often called interval 3 a ....

## An Example Of Reinforced Concrete Slab General

Various design concepts mentioned above may be illustrated considering a simple example of a reinforced concrete slab in an office building. The example shows how different design methods permissible stresses, global safety factor, partial factor method treat uncertainties of basic variables by choosing different input design values. The example also indicates significance of the reliability theory in structural design and advantages of the reliability based partial factor method compared to...

## And general three parameter lognormal distribution LNa a of basic variables X X X X X X and X

A General three-parameter lognormal distribution for anya 1. Parameter C and skewness a Distribution bound x0 - 6 a for zero a 2. Probability density lt gt and distribution function O for any a Standardised variable u x, ,a x- Transformed standardised variable sign a W ln 1 C a 2 u x, , a otherwise Density probability function o x, ,a,a pnorm uu x, ,a,a ,0, l B FORM method for determination ofthe reliability index p and probability pf Coefficients a0, a1, a2, a3, a4, a5, a6 and a7 of the limit...

## Reliability Index Beta

This notebook compute the reliability index, failure probability and influence factors in level II, using the package 'Reliability'Level2. In this package those variables are determined through the algorithm 'Normal Tail Approxima tion as is explained in the book of Madsen et al. Methods of Structural Safety, pp. 94 and following. The failure function of the limit state must be defined and, also, the independent basic variables given by a matrix with a row for each variable with the kind of...