## Appendix B Effect Of The Resistance Variability

In reliability analysis of a generic cross section the coefficient of variability VR 0.15 and the partial factor yR 1,15 are assumed as an example of a code condition. However, resistance of various structural members made of different materials may have different variability and the partial factor. The coefficient of variability VR can be expected within a broad range from 0,05 up to almost 0,50 (including uncertainty resistance model). This should be reflected by appropriate value of the...

## Appendix B Probability Updating

This example of probability updating is adopted from 4 and 5 . Consider the limit state function G(X), where X is a vector of basic variables, and the failure F is described by the inequality G(X) < 0. If the result of an inspection of the structure I is an event described by the inequality H > 0 then using equation (1) in the main text the updated probability of failure P(F I) may be written as P(F I) P(G(X) < 0 H > 0) P(G( P > 0) (B.1) For example consider a simply supported steel...

## Appendix C Bayesian Method For Fractile Estimation

Fractiles of basic variables can be effectively updated using the Bayesian approach described in ISO 12491 3 . This procedure is limited here to a normal variable X only for which the prior distribution function n' (u,c) of and cris given as n'( ,c) C o (+5(w,))expj- C- v'(sf + ' -m')2 where C is the normalising constant, 5(nr) 0 for n' 0 and 5(nr) 1 otherwise. The prior parameters m', s', n', V are parameters asymptotically given as E(u) m', E(a) s', V(u) -1 , V(a) - (C.2) while the parameters...

## 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...

## Assessment via the characteristic value

When determining the design value Xd of the parameter X from the assessed characteristic value Xk , we use the following equation according to Eurocode 1990 1 where the characteristic value Xk is given by This equation is equal to equation (6) with the coefficient of variation Vx given by The Ym is the partial factor for the parameter X and it should be taken from the appropriate Eurocode EN 1992 to EN 1998. The n is the design value of the conversion factor. This factor covers the differences...

## Background documents

Background documents related to assessment of existing structures are limited to few national codes and three International Standards ISO 2394 1 , ISO 13822 2 and ISO 12491 3 . General principles and rules of the Eurocode EN 1990 Basis of structural design 4 must be supplemented by specific procedures provided in the above mentioned International Standards ISO 1,2,3 that are primarily used in this contribution. Additional information concerning assessment of existing structures may be found in...

## Background materials

Elementary concepts and techniques of the theory of probability and mathematical statistics applicable to civil engineering are available in a number of standards 1 to 5 , background materials 6, 7, 8 , software products 9, 10, 11 and books 12 to 24 . Additional information may be found in the extensive literature listed in the books 12, 13 and others. In particular, documents developed by JCSS 6, 7 and recently published handbook 8 are closely related to the statistical techniques described in...

## Characteristic values

For verification of the structural reliability using partial factor method, the characteristic and representative values of basic variables shall be considered as follows (a) Dimensions of the structural elements shall be determined on the basis of adequate measurements. However, when the original design documentation is available and no changes in dimensions have taken place, the nominal dimensions given in the documentation may be used in the analysis. (b) Load characteristics shall be...

## Concluding Remarks

The newly available EN 1990 provides alternative design procedures and parameters that should be unambiguously specified in the National Annexes of Member States of CEN. These alternative design procedures lead in some cases to significantly different reliability levels. Preparation of National Annexes is therefore a complicated task for each Member State. Furthermore, the Eurocode standards recognise the responsibility of the regulatory authorities in each Member State and safeguard their...

## Ctexygyqyw

AE(X YG YQ YW) wE0(x,YG,YQ,YW)3-aE0(x,YG,YQ,YW) + 6-w62 WE0(X,YC,YQ,YW)2 + w63-a6 (> aE(x,YG,YQ,YW)2 + 4 + aE(x,YG,YQ,YW)) - (> aE(x,YG,YQ,YW)2 + 4 - aE(x,YG,YQ,YW) mE(x,YG,YQ,YW) -ln( c(x,YG,YQ,YW) ) + ln(crE(x,yc,yq,yw) - (0.5)-ln(1 + c(x,yc,yq,yw)) sE(x,YC,YQ,YW) yjln(1 + C(X,YC,YQ,YW)) X0X,YC,YQ,YW) E(X,YC,YQ,YW)

## 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...

## Direct assessment of the design value

When determining the design value Xd directly, we should use the following formula In case this method is used, the relevant limit states and the required level or reliability should be accounted for. The conversion factor r d should cover all uncertainties not covered by the test. The factor kd,n is obtained from the prediction method of fractile estimation with the lower value of about 0,1 (the probability p 0,001). When the coefficient of variation Vx is known, then parameter X is assumed to...

## Distribution functions

Distribution of the data points datosord > SortAdataE probab > TableB ffiMMffli, 9i, 1, ndata F ndata, 1. loglogprob > . LogA. LogAprobabEE pointsl > 99datosordQlU, probabQlU points2 > 99datosordQlU, . LogA. LogAprobabQlUEE DoApointsl > AppendToApointsl, 9datosordQiU, prcbabQiU E, 9i, 2, ndata E DoApoints2 > AppendToApoints2, 9datosordQiU, lcglcgprcbQiU E, 9i, 2, ndata E pl> ListPlotApointsl, DisplayFunction a Identity, PlotRange a AllE pp3 > ListPlotApoints2, Prolog a...

## 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...

## Example

Reinforcement cover layer of a reinforced concrete cross-section X has a mean 25 mm and standard deviation g 10 mm. The probability density function (x) for a normal distribution and for a lognormal distribution with a lower limit at zero is shown in Figure 3.2. It follows from Figure 3.2 that the normal distribution leads to occurrence of negative values of the reinforcement cover layer, which obviously does not correspond to reality. On the other hand, the lognormal distribution with lower...

## Failure probability pfPER for lognormal distribution LNjct a of E and R

Input parameters for E and R jE 50 ctE 10. aE 0,0.1 2 x 0,0.1jE 3jEwE jR 100 ctR 10. aR 1,-0.9 2 x 0,0.1jR 3jR wR - Distribution parameter C given by the skewness aE Distribution parameter CR given by the skewness aR 2. Integration bounds assuming aE> 0, aR arbitrary x0(aE,aR) max(x0EaE),x0R(aR)) if aR > 0 xC(0.608,0.301) i x1(aE,aR) jR + 6-ctR if aR > 0 x(0(0.60S, 1) x0R(aR) otherwise x1(1, 1) i 3. Transformation to the standardised normal distribution o(u) (for any a) Standardised...

## First estimate pO assuming normal distribution of E R and margin Z R E

AZ( G,yQ) -JaR (yG , Q)2 + aE2 cZ(1.2 ,1.4) 0.012 aZ( G,yQ) I pf(tG,tQ) pnorrn(- 0( G, Q) ,0,1) 0 (1.35,1.5) 3.612 index p1 assuming Gamma distribution of E and lognormal of R Gamma distribution of E k X E(x) dgamma(S, x, k) X Lognormal distribution of R having the lower limit 0 C(yG,yQ) m(7G Q) ln(cR( G, Q)) - ln( C(yG ,yQ) ) - (0.5) ln(l + C(yG,yQ)2) Rln (x, yG , yQ) plnorm (x , m (yG, yQ), s (yG, yQ)) Probability and reliability index E Gamma 1 (tG, Q) -qn rrn(pf( G, Q) ,0,1)

## Fractiles of a general three parameter lognormal distribution

Skewness a as a range variable a _1, _0.5 1 Parameter C of three s r - j 3 s r - j 3 parameter lognormal a + 4 + a) a + 4 _ aj distribution of g C(a) 1 Parameters of transformed variable mg(a) _ln( C(a) ) + ln(a) _ (0.5) - ln(1 + C(a)2) sg(a) Vln(1 + C(a)2) x0(a) -- a Check exp sign(a) u(p) - V ln (1 + C(a)2))

## G qL

The following table shows the variables that are considered in the study using the MATHEMATICA notebook FORM.nb Reliability index p 3,56 Probability of failure O(-yff) 1,87x10-4 Variable 01 As fy d fc 02 g q Sensitivity coefficient -0,383 -0,193 -0,201 -0,177 -0,018 0,761 0,300 0,274 From the results it is possible to draw the following conclusions the reliability index is a little bit lower the coefficient of influence of the concrete resistance, fc, is almost zero, it could be possible to...

## GammaR for resistance assuming normal and lognormal distribution

Annex B -Elementary methods of structural reliability II 5 GammaG for permanent load assuming normal distribution yGn( 3, V) 1 - pE(p)-V yRn(3.8,0.1) 1.2 yGn( 3, V) 1 - pE(p)-V yRn(3.8,0.1) 1.2 6 GammaQ for variable load assumingGumbel distribution dgum(3.8,0.5) 2.937 yQgum(3.8,0.5) 1.279 dgum(3.8,0.5) 2.937 yQgum(3.8,0.5) 1.279 Note. Calculation procedures applied in this sheet for determination of Gamma factors do not take into account model uncertainties of relevant variables. An additional...

## Number of samples n

Coefficient of variation Vm of the average value as a function of the number of samples. Comparison of analytical result with results from the tests. STATISTICAL DETERMINATION OF A SINGLE PROPERTY This section gives expressions for deriving design values of the ultimate resistance or serviceability parameters of a structure or a component and for deriving the design values of material properties. It is assumed that all variables follow normal or lognormal distribution and that there...

## Parametric study ofy G

Note Reliability assessment assuming normal distribution for E and R seems to be on a safe side (leads to a lower bound for p , while assessment assuming three parameter distribution for the reliability margin g seems to provide a more realistic estimate. Attachment 2 - MATHCAD sheet DesVRod.mcd

## Performs of Chi square test and draw the density function

Ej > iCDFAdistAiE, intervalsQj , luE . CDFAdistAiE, intervalsQjuEM total inj . ejM2. alfa > il . CDFAChiSquareDistributionAnumintervals . lE, estadWEM l00. alfal > il . CDFAChiSquareDistributionAnumintervals . 3e, estadWEM l00. PrintAdistAiEE PrintAEstimator W > , estadW, Degrees of freedom > , numintervals . lE PrintAConfidence level between , alfal, and , alfa, E PrintA E fdensity > PlotAPDFAdistAiE,xE, 9x, intervalsQlu , .00l, intervalsQnumintervals , DisplayFunction a IdentityE...

## 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...

## Reasons for assessment

In general, an existing structure may be subjected to the assessment of its actual reliability in case of - rehabilitation of an existing constructed facility during which new structural members are added to the existing load-carrying system - adequacy checking in order to establish whether the existing structure can resist loads associated with the anticipated change in use of the facility, operational changes or extension of its design working life - repair of an existing structure, which has...

## References

1 EN 1990 Eurocode - Basis of structural design. CEN 2002. 2 ISO 2394 General principles on reliability for structures, ISO 1998. 3 ISO 12491 1997(E) Statistical methods for quality control of building materials and components, ISO 1997. 4 JCSS Probabilistic model code. JCSS working materials, http www.jcss.ethz.ch , 2000 5 Kreyszig, E. Advanced Engineering Mathematics, John Wiley & sons, New York, Chichester, Brisbane, Toronto, Singapore, 1993.

## 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 analysis of the limit state function ZRErATzeGQ Resistance variables

Model parameter pj- 1 wr 0.1 crr wr pj Steel area mA( G , Q) A( G , Q)vA 0.05 aA( G , Q) wA pAfrG , Q) Steel strength pf 560 wf 0.05356 nf wf-pf 29.994 Internal arm pzfyG Q) zk( G, Q)wz 0.07jz( G, Q) wz pz(yG,yQ) oz(1.2,1.4) 0.015 Not directly needed Concrete area pjd d wd 0.03 crd wd pjd Action variables Concrete strenght 30 wc 0167 aC wc Model parameter - Lognorrn.d. pje ek we 0.1 ere we pje ae 3we + we Permanent Load - normal d. pG Gk wG 0.1 aG wG pG Variable Load - Gumbel d. pjQ 0.0008 wQ...

## Reliability assessment using integration

Assuming normal distribution for e Assuming gamma distribution for e Assuming lognormal distribution of R having the lower limit at a (0 default) m(yG) (ctR(yG)) - ln(C G)) - (0.5) ln(l + C(yG)2) S(yG) V ln(l + C(yG)2) Probability lognormal distribution of R Rln(x, yG) plnorm (x - a(yG)) , m(yG) , s(yG) Failure probability Prob r< e and reliability index p E has normal, R lognormal distribution En(x) Rln(x ,yG) dx n(YG) -qnorm(Pfn(YG), 0, 1) E has gamma, R lognormal distribution Eg(x) Rln(x...

## Reliability assessment without integration

Reliability index assuming normal distribution of g (a first estimate) ug(yG) p0(yG) r lLJ. Pf0(yG) pnorm(-p0(yG), 0, 1) Reliability index assuming three parameter lognormal distribution of g (a refine estimate) Parameter C of three parametr lognormal distributon of g Parameters of transformed variable mg(yG) -In(ICyG) ) + ln(ag(yG)) - (0.5) - In(1 + C(yG)2) sg(yG) > ln(1 + C(yG)2) x0(yG) ig(yG) - ag(yG) Pf1 (yG) plnorm(0 - x0(yG), mg(yG), sg(yG)) 1 (yG) -qnorm(Pf1 (yG), 0, 1) 1(1.35) 4.76

## 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...

## Reliability indexp versus ratio x imit for dominant action

A if(k < k0, 1,yQ b if(k > k0,1,yW) Limit value of x for (6.10a) and (6.10b) xx(yG,yQ,yW) yG(1 - )(1 + k) + yQ-(a - yQ) + yW-k-(b - yW) Limit of xfor (6.10a-mod) and (6.10b) xxa(yG,yQ,yw) - y& (1 - )(1 + k) + (a + yW k b) Target probability pt 3.8 Auxiliary x0 3,3.5 5 xl xx(135,15,15) Xa 0,0.05 xl + 0.05 xb xla - 0.01,xla + 0.04 0.999 xam 0,0.05 xla + 0.04 Check ln(0.999,1.15,0.15,1.35, a-1.5, b-1.5) 3.583 xc 0,0.05 xla + 0.04 ln(0,1.15,0.15,1.35,a-1.5,b-1.5) 3.591 Turkstra's for 50...

## Sample characteristics

A sample characteristic is a quantity used to describe the basic properties of a sample. The three basic sample characteristics, which are most commonly used in practical applications, are - the mean m representing the basic measure of central tendency - the variance s describing the basic measure of dispersion and - the coefficient of skewness a giving the basic measure of asymmetry. The sample mean m (an estimate of the population mean) is defined as the sum with the summation being extended...

## The General Case Of Reliability Analysis General

In the Chapter II Elementary methods of structural reliability I of this handbook it has been presented what is called the fundamental case of structural reliability. It is the case in which the limit state function can be represented by only two random independent variables, the effect of the actions and the resistance. This fundamental case represents a very interesting case for introducing the reliability concepts due to its very intuitive reasoning, and that we are used to this concepts of...

## Theoretical value of the global factory for a given exceedance probability p of Ed

P P E > Ed p 0.001,0.0011 0.006 Probability considered in EN pp (Ed, x , yG, yQ, yW) plnorm(Ed - x0(x, yG, yQ, yW), mE(x, yG, yQ, yW), sE(x, yG, yQ, yW)) Ed(p,x,yG,yQ,yW) x0(x,yG,yQ,yW) + qlnorm(1 - p,mE(x,yG,yQ,yW),sE(x,yG,yQ,yW)) Ek(x,yG,yQ,yW) Gx,yG,yQ,yW) + Qk(x,yG,yQ,yW) + Wk(x,yG,yQ,yW) yp(0.002,0.3,1.5,1.5,1.5) 1.228 8 The global load factory versus ratio x limit for dominant action ko - Auxiliary quantities a if(k < kO, 1, yQ) Limit value of x for (6.10a) and (6.10b) xx((,yQ,yW)...

## Three parameter lognormal distribution of E

AE0(x yG yQ yW) oQ(x,yG,yQ,yW) aQ + CTW(x,yG,yQ,yW) aW 3 2 2 3 i wE0(x,yG,yQ,yW)aE0(x,yG,yQ,yW) + 6 w9 wE0(x,yG,yQ,yW) + w9 a9 aE(x,yG,yQ, yW) - aE(x, yG, yQ, yW) + 4 + aE(x, yG, yQ, yW) mEx,yG,yQ,yW) -ln(c(x,yG,yQ,yW)) + ln(< 7E(x,yG,yQ,yW)) - (0.5)-ln(1 + C(x,yG,yQ,yW)2) sE(x,yG,yQ,yW) ln(1 + c(x,yG,yQ,yW)2) x0(x,yG,yQ,yW) E(x,yG,yQ,yW) Probability density of E, approximation by three parameter lognormal distribution xC(0.4,1.35,1.5,1.5) 0.189 Eln(x, x,yG, yQ,yW) dlnorm(x - x0(x,yG,yQ,yW),...

## Updating of characteristic and design values

The updating procedure (2) can be used to derive updated characteristic and representative values (fractiles of appropriate distributions) of basic variables to be used in the partial factor method. The Bayesian method for fractile updating is described in Annex C to this Chapter. More information on updating may be found in ISO 12491 3 . A more practical procedure is to determine directly updated design values for each basic variable. For a resistance parameter X, the design value can be...

## 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...

## Iso 2394 And Eurcode

1 EN 1990 Eurocode - Basis of structural design. CEN 2002. 2 ISO 2394 General principles on reliability for structures, ISO 1998. 3 JCSS Background documentation, Part 1 of EC 1 Basis of design, 1996. 4 Gulvanessian, H. - Calgaro, J.-A. - Holicky, M. Designer's Guide to EN 1990, Eurocode Basis of Structural Design Thomas Telford, London, 2002, ISBN 0 7277 3011 8, 192 pp. 5 JCSS Probabilistic model code. JCSS working materials, http www.jcss.ethz.ch , 2001. 6 Melchers R.E. Structural...

## 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...

## Design reliability parametric study of yG yQ Applied units m MN MPa

Characteristic actions Gk 0.007 Qk 0.003 Material strength fk 500 ck 20 Material factors - ic 1.5 rn 1.15 For the global factor method 7c 7m 1.0 Load factors Prarneters yG 1 ,1.05 1.5 yQ 1 ,1.05 1.6 Global factor if any s0 1.0 Load effect L 6 ME G sO ek yG Gk Q Qk ME 1.35,1.5 0.063 Cross-section b 1 d 0.25 - 0.03 Choose reinfocernent e.g. A yG,yQ 0.00204 If not, than A A yG ,yQ b d If yes, put the red box on this line gt Reinforcement area cm2 DOOO A 1.35,1.5 6.917 inforcement ratio p 1.35,1.5...

## 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...

## Hazard Identification

A hazard is a set of circumstances, possibly occurring within a given system, with the potential for causing events with undesirable consequences. For instance the hazard of a civil engineering system may be a set of circumstances with the potential to an abnormal action e.g. fire, explosion or environmental influence flooding, tornado and or insufficient strength or resistance or excessive deviation from intended dimensions. In the case of a chemical substance, the hazard may be a set of...

## 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

## Design values Ed and Rd

EC 1990 recommendation p 3.8 Ed yG E - aE yG p ctE Ed0 yG E - aE0p ctE RdOin yG R yG exp -aR0p vR Check Ed 1.35 1.22 Ed0 1.35 1.27 Rd yG R yG - aR yG p ctR yG Rd0 yG R yG - aR0p ctR yG Ed yG 1.6 Rd yG Ed0 yG Rd0 yG Rd0in yG Ed yG 1.6 Rd yG Ed0 yG Rd0 yG Rd0in yG Notes 1 Figure shows that the partial factor y g should be greater than about 1,25 otherwise the design value of the load effectEd would be greater than the design value of the resistance Rd. 2 The design value of the resistance Rd...

## 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...

## Ae

Where symbols g, e and w are defined in Table 1. However, the concept of the Implied Cost of Averting a Fatality described by equation 6 is just one of possible approaches to the complex problem of evaluating social consequences. At present further intensive investigation is expected. Risk is commonly estimated by the mathematical expectation of the consequences of an undesired event that often leads to the product probability x consequences. As a rule the risk of civil engineering systems is a...

## 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...

## Sensitivity factors

Sensitivity factors of the First Order Reliability Methods FORM are normally used 1,2 to calibrate design values of basic variables and partial safety factors. Considering the limit state function Z X reliability margin given by equation 11 , the sensitivity factors for the four cumulative variables R, G, Q, W can be defined in terms of their standard deviations or, og, lt jq and oW as follows c r 0 , amp G 0 , aQ 0 , aw 21 where og denotes the standard deviation of Z X given as In the...

## Fundamental Load Combinations

In the following, the combination of three actions is considered permanent action G, imposed load Q leading and wind W accompanying . EN 1990 1 for the fundamental combination of these loads in persistent and transient design situations introduces three alternative procedures denoted here A, B and C. The loads actions G, Q and W and their characteristic values Gk, Qk and Wk denote generally load effects for example internal bending moments of appropriate loads actions and should be...

## 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...