Fit data to distribution functions
Statistical characterization of data. It takes the following steps 1- Obtain the characteristics of datas number of elements, mean, variance, standard deviation and coefficeint of variation 2- By the method of moments obtain the indicated distribution parameters 3-Manually the values of the steps in the hystogram are introduced, and then perform the Chi square test of goodness of lit and draw the hystogram and the adjusted density distribution function. Gives two values of Confidence level, the...
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
Figure 4.2. 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 Qde
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 1
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
Let X be lognormaly distributed. 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...
Ft
Figure 5. Experimental Fe vs. theoretical Ft resistance. The mean and sample standard deviation of the logarithm of the error term 8 are given by A 0,000567, sA 0,000277 , the coefficients of variation are calculated from equations 36 , 40 and 43 and are V82 0,000277, Vrt2 0,000236, Vr2 0,000513. The factor kn 1.78054, is obtained using the approximation formula. The theoretical value of the resistance function at the mean values of basic variables is rm 252,5 kN. Finally, the characteristic...
Tail sensitivity problem
In statistics, the decisions are, generally, not taken after a mathematical proof'. A hypothesis is accepted if evidence for its rejection is not found. The probability distribution assigned to any variable can have an important influence on the estimated probabilities of failure. Besides, when assigning distribution functions, the data available are mainly, of course, of the central part of the distribution where the election of one or other distribution function is no very significant. This...
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...
Chapter I 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...
Example Yqh
A sample of n 5 concrete strength measurements having the mean m 29,2 MPa and standard deviation s 4,6 MPa is to be used to assess the characteristic value of the concrete strength fck xp, where p 0,05. If no prior information is available, then n' V 0 and the characteristics m, n, s, V equal the sample characteristics m, n, s, v. The predictive value of xp then follows from C.5 as frayes 23,9 - 1,8 X 4,3 X - 1 15,5 MPa where the value tp - 2.13 is taken from Table C.1 for 1 - p 0.95 and v 5 -...
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...
g q L
L denotes the span of the simply supported beam. Using equilibrium conditions 29 and reinforcement area As can be derived Without going into technical details note, that equation 33 may be used approximately for the global safety factor methods and partial safety factor method. Attached EXCEL sheet RCBeam and MATHCAD sheet RCBeam may be applied to make necessary calculations. The classical permissible stresses method assumes a triangular compressive stress block in the compressive concrete...
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 ....
Partial factor or global safety factor only
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 As Md z fyd A mA2 P P gt pmin x d lt max General Table acc 1 fcd acc fck yc 13,3 fyd fyk ya 434,8
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...
References Qwd
1 EN 1990 Eurocode - Basis of structural design. CEN 2002. 2 ISO 2394 General principles on reliability for structures, ISO 1998. 3 ISO 13822. Basis for design of structures - Assessment of existing structures, ISO 2001. 4 JCSS Probabilistic model code. JCSS working materials, http IIwww.jcss.ethz.chI, 2001. 5 Gulvanessian, H. - Calgaro, J.-A. - Holicky, M. Designer's Guide to EN 1990, Eurocode Basis of Structural Design Thomas Telford, London, 2002, ISBN 07277 3011 8, 192 p. 6 JCSS Background...
and general three parameter lognormal distribution LNa a of basic variables X 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...