NPTEL STRUCTURAL RELIABILITY

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1 NPTEL Course O STRUCTURAL RELIABILITY Module # 0 Lecture 1 Course Format: Web Istructor: Dr. Aruasis Chakraborty Departmet of Civil Egieerig Idia Istitute of Techology Guwahati

2 1. Lecture 01: Basic Statistics Scatter Diagram, Histogram ad Frequecy Polygo Observatio data samples are preseted i form of scattered poits which ca be idepedet or depedet of ay other radom variable. Presetatio of the sample data is vitally importat as it gives crucial kowledge about its costitutive statistical properties such as correlatio, rage etc. Geerally, a statistical observatio sample is represeted i scatter diagrams, histograms ad frequecy polygos. The radom variables associated with these observatios are discrete. Geerally, scatter diagram is preseted either i D or 3D form by presetig two or three radom variables, respectively. Figure.1.1 shows typical scatter diagrams for two radom variables. Each radom variable must have observatio data which ca be discretely represeted across graph. Thus, the statistical data must be relatig to simultaeous measuremet of the radom variables. The scatter diagrams shows ature ad relatio of the radom variables with each other. For example, if two radom variables have icreasig tred i scatter diagram that meas they have positive correlatio ad vice versa [see Figure.1.1 (a) ad (b)] whereas if this icreasig or decreasig tred is very strict (i.e. early followig a straight lie), oe ca say that the correlatio is either +1 or 1, respectively [see Figure.1.1 (e) ad (f)]. At times oe ca otice that treds i scatter diagram is either uiformly icreasig or decreasig, these have early zero liear correlatio [see Figure.1.1 (c) ad (d)] whereas a strog quadratic correlatio exits betwee the pair of radom variables show i Figure.1.1 (d). Histograms are represetatio of grouped frequecy distributio of observatio data. These are bar like represetatio of observatio data where width of bar is class iterval of the data ad amplitude or height of bar refers to frequecy desity of data fallig uder its associated class (see Figure.1.). The area of each bar represets its class frequecy, this is expressed i Eq Area of each rectagle = width height = (width of class) (frequecy desity) = (width of class) = class frequecy class frequecy width of class.1.1 Course Istructor: Dr. Aruasis Chakraborty 1

3 Frequecy Lecture 01: Basic Statistics y y y x x x (a) (b) (c) y y y (d) x (e) x (f) x Figure.1.1 Scatter diagrams showig differet types ad degrees of correlatio (a) positive, b egative, (c) zero, d zero, e +1 ad f 1 Histogram Frequecy Polygo Values Figure.1. Typical example of histogram ad frequecy polygo Course Istructor: Dr. Aruasis Chakraborty

4 Lecture 01: Basic Statistics Before plottig histograms oe has to form frequecy table which cotais class ad frequecy. Choosig umber of classes play a very crucial role i formulatio of frequecy table, i tur histograms. Geerally, appropriate umber of classes may be chose by usig c = log.1. where, is umber of observatio data or sample size ad c is umber of classes. A alterative to histograms is frequecy polygo which formed by joiig the mid values of each class as show i Figure.1.. If the width of class are same tha the area uder histograms is same as uder the frequecy polygo. The curve formed by frequecy polygo gives a idea of frequecy distributio of the data. Measures of Cetral Tedecy A whole set of observatios ca be described by a sigle value. It usually occupies a cetral positio such that some observatios are larger ad some others are smaller tha itself, these are kow as measures of cetral tedecies. There are 3 measures of cetral tedecy mea, media ad mode. Mea It is of 3 types arithmetic mea, geometric mea ad harmoic mea. The words 'mea' ad 'average' oly refer to arithmetic mea. I this course oly arithmetic mea is discussed. Arithmetic Mea (AM) It is defied as sum of a set of observatios divided by size of the set. Cosider observatios x 1, x,, x where is umber of observatios, their AM (μ x ) is μ x = x 1 + x + + x = 1 x.1.3 Now, say x 1, x,, x have frequecies f 1, f,, f respectively, i.e. x 1 occurs f 1 times, x occurs f times ad so o, the the sum of all the observatios (i.e., f 1 + f + + f ) is x 1 + x x 1 f 1 terms + x + x + + x f terms + + x + x + + x f terms = f 1 x 1 + f x + + f x.1.4 Hece, the arithmetic mea is μ x = f 1x 1 + f x + + f x f 1 + f + + f = fx f.1.5 Course Istructor: Dr. Aruasis Chakraborty 3

5 This is sometimes referred to as weighted arithmetic mea. Lecture 01: Basic Statistics Importat properties of AM 1. Additio of a set of observatios is equal to the product of umber of observatios ad AM. x i = μ x ad f i x i = Nμ x.1.6 where N = f is the total frequecy. The first relatio i Eq..1.6 implies that the simple sum whereas the secod relatio implies the weighted sum.. For give observatios, the sum of deviatios from their mea is always 0. x i μ x = 0, f i x i μ x = 0, where μ x = x i, ad where μ x = f ix i N Two variables x ad y, related i such a way that y = ax + b, where a ad b are costats, the μ y = aμ x + b.1.8 ad vice versa. Relatio i Eq..1.8 explais that if each of the observatios x i is added, subtracted, multiplied or divided by a costat tha the mea μ x will also follow the same mathematical operatio ad that too with same costats. 4. Let a group of two observatios of size 1 ad havig meas μ x1 ad μ x, the the combied mea (μ x ) of the composite group of 1 + (= N) observatios is give by Nμ x = 1 μ x1 + μ x.1.9 This ca be geeralised to ay umber of groups as Nμ x = i μ xi where N = i The sum of squares of deviatios has the smallest value if deviatios are take from their mea or AM. x i A is miimum, whe A = simple AM.1.11 Course Istructor: Dr. Aruasis Chakraborty 4

6 f i x i A is miimum, whe A = weighted AM Lecture 01: Basic Statistics Media The middle most value whe a set of observatios are sorted i order of magitude is called media. It ca be calculated from a grouped frequecy distributio by usig the formula : Media = l 1 + N/ F f m c.1.1 where, l 1 is lower boud of the media class, N is total frequecy, F is cumulative frequecy correspodig to l 1, f m is frequecy of the media class ad c is width of the media class. Media is, i a certai sese, the real measure of cetral tedecy because it gives the value of the most cetral observatio. Moreover, it is uaffected by higher or lower boud values, ad ca be easily calculated from frequecy distributios with ope-ed classes. Mode The value i a set of observatios which occurs with the highest frequecy is kow as mode. This actually, reflects the most ofte occurrig value. It is geerally calculated as Mode = l 1 + d 1 d 1 + d c.1.13 where, l 1 is lower boud of the highest frequecy class, d 1 is differece of frequecies i the highest frequecy class ad the precedig class, d is differece of frequecies i the highest frequecy class ad the followig class, ad c is commo width of classes. Eq..1.3 is applicable oly whe all classes have the same width. Oe ca ote that mode has a peculiarity, i.e., i case of observatios occurrig with equal frequecy, mode does ot exist. Relatio Betwee Mea, Media ad Mode A iterestig approximate empirical relatioship betwee mea, mode ad media exist ad it ca be expressed as Mea Mode 3(Mea Media).1.14 Note: this expressio oly holds fairly for sigle mode with moderate asymmetry. Stadard Deviatio ad Variace Variace is defied as arithmetic mea of squared deviatio from mea, where the deviatio from mea, square deviatio from mea ad variace are show below Deviatios from mea: x 1 μ x1, x μ x,, x μ x.1.15 Course Istructor: Dr. Aruasis Chakraborty 5

7 Square-Deviatios from mea: x 1 μ x1, x μ x,, x μ x Mea-Square-Deviatios from mea: Lecture 01: Basic Statistics 1 x 1 μ x1 + x μ x + + x μ x = 1 x i μ xi Variace is geerally deoted by σ, further, below expressios for simple series as well as frequecy distributio are give. For simple series, σ = 1 x i μ xi.1.16 For frequecy distributio, σ = 1 N f i x i μ xi.1.17 Stadard deviatio, σ is defied as square root of variace. It is evaluated as show i Eq Stadard Deviatio (σ) = 1 x i μ xi.1.18 Both, variace ad stadard deviatio are vital tools for represetatio of a statistical data as it shows dispersio of the data from mea i its domai. Covariace ad Correlatio Coefficiet Covariace is defied for pair of radom variables which is associated or related to each other. It is the average of product of idividual deviatio from the correspodig meas. Eq shows covariace Cov x, y betwee two correlated radom variables x ad y. Cov x, y = 1 x i μ xi y i μ yi.1.19 Expadig Eq..1.19, oe ca get Cov x, y = xy x y.1.0 Course Istructor: Dr. Aruasis Chakraborty 6

8 Lecture 01: Basic Statistics Geerally, oe expresses the correlatio of two radom variables i terms of coefficiet of correlatio (ρ) which is the ratio of covariace ad idividual stadard deviatios of both the radom variables. ρ = Cov x, y σ x σ y.1.1 Substitutig the values of Cov x, y, σ x ad σ y from Eq ad.1.18 i Eq..1.1, oe gets ρ = x μ x y μ y x μ x. y μ y.1. Expadig Eq..1., ρ = xy μ x μ y x μ x y μ y.1.3 As μ x = x ad μ y = y, oe ca substitute this to the above equatio ad o simplifyig, ρ = xy x y x x y y.1.4 Percetile Percetile is a value below which a give percetage of observatios fall. For example, 99% of the observatios will fall uder 99 percetile (P 99 ). As per rak the values of differet percetiles ca be arraged as P 1 < P < < P 99. Regressio Regressio is a estimatio process doe for average value of oe variable for a specified value of other variable. It is coducted with respect to suitable equatios (i.e., regressio equatios) based o statistical data (combied as well as idividual) of the radom variables. For simple regressio, oe ca cosider liear relatioship betwee the variables. Hece, estimates of y (deoted by y ) is give by regressio equatio of y o x as Course Istructor: Dr. Aruasis Chakraborty 7

9 Lecture 01: Basic Statistics y μ y = b yx x μ x.1.5 where, regressio coefficiet b yx = Cov x, y σ x ad similarly, regressio equatio of x o y is give as Eq..1.6 for estimate of x (deoted by x ) x μ x = b xy y μ y.1.6 where, regressio coefficiet b xy = Cov x, y σ y. Now cosider a straight lie fit as show below for better uderstadig of formulatio ad calculatios related to regressio. y = a + bx.1.7 where, radom variable x is idepedet whereas y is depedet of x. Hece, i Eq..1.7 oe gets coefficiets a ad b as ukow terms which are to be evaluated as per regressio. Multiplyig Eq..1.7 by 1 ad x, moreover summig up the observatios of the radom variables, oe gets y = a + b x.1.8 xy = a x + b x.1.9 Cosiderig Eq..1.8, dividig by (umber of observatios) oe gets μ y = a + bμ y.1.30 a = μ y bμ x.1.31 thus, ukow coefficiet a is evaluated i terms of idividual mea of both the radom variables. Now, multiply Eq..1.8 by x ad divide Eq..1.9 by Fially, subtractig Eq ad Eq..1.3 x y = a x + b x.1.3 xy = a x + b x.1.33 xy x y = b x + x.1.34 b = xy x y x + x.1.35 Course Istructor: Dr. Aruasis Chakraborty 8

10 Dividig Eq by b = xy x x + x y = cov x, y σ x Lecture 01: Basic Statistics = ρ σ y σ x.1.36 thus, aother ukow coefficiet b is evaluated i terms of covariace or coefficiet of correlatio ad variace. Substitutig a from Eq ad b from Eq oe gets similar expressio as Eq y μ y = b yx x μ x.1.37 Course Istructor: Dr. Aruasis Chakraborty 9

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