NPTEL STRUCTURAL RELIABILITY

Transcription

1 NTEL Course On STRUCTURL RELIBILITY Module # 02 Leture 2 Course Format: Web Instrutor: Dr. runasis Chakraborty Department of Civil Engineering Indian Institute of Tehnology Guwahati

2 2. Leture 02: Theory of robability Theory of robability ording to definition, probability is an outome whih ours T times in N mutually exlusive, equally likely and exhaustive trials then probability of ourrene of is given by T N i.e. relative frequeny of ourrene. Note that is an event in sample spae S. Three axioms of probability are given by xiom I: robability of ourrene of any event an neither be less than 0 nor be greater than 1, i.e xiom II: The ertainty of outome is unity i.e. S xiom III: Outomes are mutually exlusive, equally likely and exhaustive. So, if mutually exlusive events in S, then 1, 2,... are For finite number of suh events (say k ), k r k s an example, onsider the design of a struture. fter onstrution only two outomes are possible either suess or failure. Both are mutually exlusive, they are also alled exhaustive and no other outome is also possible. r1 1

3 Leture 02: Theory of robability failure 1 suess The probability of suess of the struture is Reliability, whih is given by R f 1 or R f Set Theory Let us onsider throwing of a die. The result of eah throw is a number from 1 to 6. Eah throwing is an event and eah result is an outome. Colletion of all possible outomes is alled sample spae, whih is normally defined by S. Note that sample spae may be finite or infinite. problem for readers to be solved by themselves. Considering the ase of die outomes, as explained above and find out 2 4., Events an be defined in various types, as Simple Event : that onsists of only one event Compound Event : made up of two or more simple event Certain Event : S an event that onsists of all possible sample point in the sample spae Null Event : omplement of ertain event 1,2,3,4,5,6 Venn Diagram Sets or events, as disussed above under Set Theory, an be expressed with logially relations between them in the sample spae via a speial format of diagram alled as Venn diagram. This was first introdued by British mathematiian, John Venn, in year Venn diagrams are widely aepted and are easy to represent. The diagram is bounded in a retangular box whih, in fat, represents the sample spae of the sets, as shown in Figure and The sets or events an be shown in irular or any appropriate shape within the irle for logial representation of their orresponding relations between the other sets or events. Thus, relations like union ( ), intersetion ( ), mutually exlusive, omplement (. ) et. an be expressed. The figures shown below shows a set or event and its omplement (see Figure 2.2.1) and union of two sets or events (see Figure 2.2.2). 2

4 Leture 02: Theory of robability S B Figure Venn diagram showing a set and its omplement Figure Venn diagram showing intersetion of two sets few of the basi laws used in set theory are listed below: Identity Laws Idempotent Laws Complement Laws Commutative Laws DeMorgan's Laws ssoiative Laws Distributive Laws S S S S B B B B B B B B B C BC B C BC B C BC B C BC where, ϕ stands for null event. Exerise (for pratie) 1. What is the probability of getting Head in a single toss of an unbiased oin? 2. die is tossed and the number of points appearing on the uppermost fae is observed. What is the probability of obtaining (a) an even number, (b) an odd number) () less than 3 and (d) a six? 3. If 10 persons are arranged at random (i) in a line (ii) in a irle, find out the probability that 2 partiular persons will be next to eah other? 4. Two vehiles are approahing a road juntion. robability of leading vehile turning right is 0.3 and that of following vehile is 0.6. The probability of both the vehiles turning 3

5 Leture 02: Theory of robability right is 0.1. Find out onditional probability that the following will turn right if the leading vehile turns right. What is the probability of the following vehile not turning right when the leading vehile is not turning right? 5. Using Venn diagram, proof that B B B B C B C B B C C B C