Stochastic Structural Dynamics

Transcription

1 Stochastc Structural Dynamcs Lecture-1 Defnton of probablty measure and condtonal probablty Dr C S Manohar Department of Cvl Engneerng Professor of Structural Engneerng Indan Insttute of Scence angalore Inda manohar@cvl.sc.ernet.n 1

2 Ths Lecture What s ths course about? egn revewng theory of probablty

3 Loads on engneerng structures Earthquake Wnd Waves Gudeway unevenness Traffc Dynamc Random

4 Loads on engneerng structures Earthquake

5 Loads on engneerng structures Wnd

6 Loads on engneerng structures Gudeway unevenness

7 Uncertantes n structural engneerng problems Loads (earthquakes, wnd, waves, gude way unevenness ) Structural propertes (elastc constants, mass, dampng, strength, boundary condtons, jonts ) Modelng (analytcal, computatonal and expermental) Condton assessment n exstng structures Human errors 7

8 Stochastc structural dynamcs ranch of structural dynamcs n whch the uncertantes n loads are quantfed mathematcally usng theory of probablty, random processes and statstcs. Random vbraton analyss; probablstc structural dynamcs Falure of structures under uncertan dynamc loads Desgn of structures under uncertan dynamc loads lso mportant n expermental vbraton analyss Measurement of frequency response and mpulse response functons Sesmc qualfcaton testng Condton assessment of exstng structures Pre-requstes asc background n Lnear vbraton analyss Probablty and statstcs

9 Mathematcal models for uncertanty: Probablty, random varables, random processes, statstcs. Fuzzy logc. Interval algebra. Convex models.

10 Revew of probablty and random processes Suggested books 1. Papouls and S U Plla, 2006, Probablty, random varables and stochastc processes, 4 th Edton, McGraw Hll, oston. 2. J R enjamn and C Cornell, 1970, Probablty, statstcs, and decson for cvl engneers, McGraw Hll ook Company, oston. Defntons of probablty: 1.Classcal defnton 2.Relatve frequency 3.xomatc 10

11 Classcal (mathematcal or a pror) defnton: If a random experment can result n n outcomes, such that these outcomes are equally lkely mutually exclusve collectvely exhaustve and, f out of these n outcomes, m are favourable to the occurrence of an event, then the probablty of the event s gven by P()=m/n. Example: P(gettng even number on de tossng)=3/6=1/2. Objectons What s equally lkely? What f not equally lkely? (what s the probablty that sun would rse tomorrow?) No room for expermentaton. Probablty s requred to be a ratonal number. 11

12 Relatve frequency (posteror) defnton If a random experment has been performed n number of tmes and f m outcomes are favorable to event, then the probablty of event s gven by lm. P n m n Objectons What s meant by lmt here? One cannot talk about probablty wthout conductng an experment. What s the probablty that someone meets wth an accdent tomorrow? Probablty s requred to be a ratonal number. 12

13 Example: Toss a de 1000 tmes; note down how many tmes an even number turns up (say, 548). P(even number)=548/1000. N=1000 here s deemed to be suffcently large. There s no guarantee that as the number of trals ncreases, the probablty would converge. The de need not be far. 13

14 xomatc defnton Undefned notons (prmtves) Experments Trals Outcomes n experment s a physcal phenomenon that can be observed repeatedly. sngle performance of an experment s a tral. The observaton made on a tral s ts outcome. xoms are statements that are commensurate wth our experence. No proofs exst. ll truths are relatve to the accepted axoms. 14

15 Random experment (E) s an experment such that the outcome of a specfc tral cannot be predcted, and t s possble to predct all possble outcomes of any tral. Remarks E : the frst techncal term. Example: Toss a con. We know that we wll ether get head or tal. In any gven tral however we do not know before hand what would be the outcome. What cannot be envsaged, does not enter the theory. 15

16 xomatc defnton (contnued) Sample space ( ) Set of all possble outcomes of a random experment. Examples (1) Con tossng: = h t ; Cardnalty=2; fnte sample space. (2) De tossng: = ; Cardnalty=6; fnte sample space. (3) De tossng tll head appears for the frst tme: = h th tth ttth tttth ; Cardnalty= ; countably nfnte sample space. (4) Maxmum ranfall n a year: = 0 X< ; Cardnalty= ; uncountably nfnte sample space. 16

17 Elements of Ω are called sample ponts. Ω can be thought of as outcome space. Consder a set wth n elements. n n n n n n Co C1 C2 Cn Number of subsets= (1 1) 2 17

18 xomatc defnton (contnued) Event space ( ) Ω s fnte : s the set of all subsets of Ω. h t; h t Ex : Ω Elements of are N Cardnalty of 2 ; N cardnalty of Ω. known as events In general: s the sgma algebra of subsets of Ω Defnton Let C be a class of subsets of Ω. If C (a),& (b) Ω Ω 1 1 then, we say that C s a sgma algebra of subsets of Ω. 18

19 xomatc defnton (contnued) Probablty (P) P : 0,1 Ω j P P, 1 such that xom1(axom of non - negatvty) P xom 2 (axom of normalzaton) P xom 3 (axom of addtvty) j Probablty space s the ordered trplet : Ω,, P 19

20 Note: We wsh to assgn probablty to not only to elementary events (elements of sample space) but also to compound events (subsets of sample space). When sample space s not fnte, ( as when t s the real lne) there exsts subsets of sample space whch cannot be expressed as countable unon and ntersectons of ntervals. On such events we wll not be able to assgn probabltes consstent wth the thrd axom. To overcome ths dffculty we exclude these events from the event space.

21 Remarks (3) P (4) P c 1 P (1) P (2) P P P 0 P P 21

22 Proof 1 E P E c c ; E E c c E E P E P E P PE P c E P c E 1 PE (xom 3) 1(xom 2) Proof 4 : Use proof c Note :. 1 wth E ;

23 P P P P P P P P P P c c c c c c and (2) From (1) (2) (xom 3) ; (1) (xom 3) ; 3 Proof 3. :Use proof 2 : hnt Proof

24 P Condtonal probablty and stochastc ndependence Defnton Probablty of event gven that has occurred P P ; P Example: Far De tossng 0. = The de has been tossed and an even number has been observed. P 2 Even? pproach 1: Even= P 2 Even 1/ 3 (Classcal defnton) pproach 2: P2 Even P2 Even PEven 2Even (2) 1/6 P2 Even 1/3. 1/2 Condtonal Probablty obeys all the axoms (1) P 0 (2) P 1 (3) P C P P C f C 24

25 Stochastc ndependence Events and are sad to be stochastcally ndependent f any one of the followng four statements s true: (1) The probablty of occurrence of event s not affected by the occurrence of event. (2) P PP (3) P P Notaton : and are ndependent P PP P (4) P ( ); P ( ) 0. P ( ) Remarks (1) Defnton 1 s not useful to verfy f and are ndependent. (2) If we need to verfy f and are ndependent, we need to fnd P, P ( ), P ( ),& P ( ) and use defntons 2,3, or 4. (3) Independence of more than two events can also be defned. Thus 3 =1 are sad to be ndependent f j j (1) P P P, j 1, 2,3 & j, and (2) P P P P

26 Example Toss two cons. 1 2 = hh ht th tt Let ab, 0, such that ( ab) Let Phh ( ) a Ptt b P ht Pth ab. Clearly P( )= P( hh) P tt P ht P th ( a b) 1. Defne two events head on the frst cont= head on the second cont= hh th E hh ht E Queston : verfy f E & E are ndependent ( ) PE E PE PE P E P hh ht a aba( ab) a PE ( ) Phh th a abaa ( b) a P E E P hh a E & E are ndependent

27 Example: n whch three events are parwse ndependent but are not ndependent. Consder a far tetrahedron (ths has four faces) Let the four faces be panted as Green, Yellow, lack and G+Y PY ( ) ; PG ( ) P ( ) ; PGY ( ) PGPY ( ) ( ) 4 1 PG ( ) PGP ( ) ( ) 4 1 PY ( ) PY ( ) P ( ) PGY ( ) PGPY ( ) ( ) P ( ). 4 8

28 Example Consder a random experment nvolvng tossng of C two des. Defne (even on de1) (even on de 2) (sum of Examne f numbers on de1and de 2),, and C are ndependent.

29 Total probablty theorem 1 N Let consttute a partton of. 1 N That s, ; j. 1 Let be a set. N N 1 j N N P P P 1 1 P P P( ) 29

30 ayes' theorem P P 1 P P P( ) P ( ) P ( ) N P P( ) P P( ) P ( ) a pror probablty P ( ) posteror probablty