Simulation and Probability Distribution

Transcription

1 CHAPTER Probablty, Statstcs, and Relablty for Engneers and Scentsts Second Edton PROBABILIT DISTRIBUTION FOR CONTINUOUS RANDOM VARIABLES A. J. Clark School of Engneerng Department of Cvl and Envronmental Engneerng 5d Probablty and Statstcs for Cvl Engneers Department of Cvl and Envronmental Engneerng Unversty of Maryland, College Park CHAPMAN HALL/CRC CHAPTER 5d. PROBABILIT DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slde No. Need for Smulaton Estmatng the probablty of falure for both explct or mplct lmt state functons wthout knowng analytcal technques such as the FORM method. Smulaton provdes an unque opportunty to understand several mportant elements related to probablty dstrbutons and probablstc analyss.

2 CHAPTER 5d. PROBABILIT DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slde No. Need for Smulaton (cont d) Smulaton s used to verfy the accuracy of structural relablty methods wth lttle background n probablty and statstcs. Measured data are often very lmted, and makng decson wth small sample szes ncreases the rsk of ncorrect decson. CHAPTER 5d. PROBABILIT DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slde No. 3 Monte Carlo Smulaton Monte Carlo smulaton has sx essental elements:. Defnng the problem n terms of all the random varables,. Quantfyng the probablstc characterstcs of all the random varables (.e., mean, COV, dstrbuton type), 3. Generatng the values of these random varables

3 CHAPTER 5d. PROBABILIT DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slde No. 4 Monte Carlo Smulaton (cont d) 4. Evaluatng the problem determnstcally for each set of all the random varables, 5. Extractng probablstc nformaton from n observatons. 6. Determnng the accuracy and effcency of the smulaton. CHAPTER 5d. PROBABILIT DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slde No. 5 Formulaton of the Problem Consder a smply supported beam as shown w P L/ L 3

4 CHAPTER 5d. PROBABILIT DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slde No. 6 Assume both w and P are random varables. Thus, the desgn bendng moment M at the mdspan of the beam s also a random varable. The task now s to evaluate the probablstc characterstcs of the desgn bendng moment usng smulaton. CHAPTER 5d. PROBABILIT DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slde No. 7 If the span of the beam s 30 feet, the expresson for the desgn moment can wrtten as wl PL M = =.5w + 7.5P W and P n ths case are called basc random varables. 4

5 CHAPTER 5d. PROBABILIT DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slde No. 8 Generaton of Random Varables Computer software packages are avalable (e.g., Excel, Quattro Pro, etc.) The generated random numbers from these packages are called pseudo random numbers These numbers are generated from a welldefned and predctable process CHAPTER 5d. PROBABILIT DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slde No. 9 Mdsquare Method Ths method llustrates the problems assocated wth determnstc procedures The general procedure s as follows:. Select at random a four-dgt number (seed). Square the number and wrte the square as an eghtdgt number usng precedng (lead) zeros f necessary 3. Use the four dgts n the mddle as the new random number. 4. Repeat steps and 3 to generate as many numbers as necessary 5

6 CHAPTER 5d. PROBABILIT DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slde No. 0 Example : Mdsquare Method Consder the seed number 89. Ths value would produce the followng: CHAPTER 5d. PROBABILIT DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slde No. Transformaton of Unform Random Numbers The Unform 0 for x < 0 x a F ( x) = for a x b b a for x > b where a < b. The mean and varance are gven by µ a + b = and σ = ( b a) 6

7 CHAPTER 5d. PROBABILIT DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slde No. Inverse Transformaton Technque or Inverse CDF Method In the n nverse transformaton technque or nverse CDF method, the CDF of the random varable s equated to the generated random number u, that s, F (x ) = u, and the equaton can be solved for x as follows: x = F ( u ) CHAPTER 5d. PROBABILIT DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slde No. 3 F U (u ) F (x ) U f U (u ) f (x ) x = F ( u ) U u 0 x 7

8 CHAPTER 5d. PROBABILIT DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slde No. 4 Example: Normal If s normally dstrbuted, that s ~ N(µ,σ ), then Z = (- µ )/σ s a standard normal varate, that s, Z ~ N(0,). It can be shown that x µ u = F ( x ) = Φ( z ) = Φ σ or Thus, x x µ z = σ = µ + σ z = µ + σ Φ ( u ) CHAPTER 5d. PROBABILIT DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slde No. 5 Example: Lognormal If s lognormally dstrbuted, that s ~ LN(µ,σ ), then Z = (ln- µ )/σ s a standard normal varate, that s, Z ~ N(0,). It can be shown that ln x µ u = F ( x ) = Φ( z ) = Φ σ or Thus, ln ( x ) = µ + σ Φ ( u ) x = e [ µ + σ Φ ( u )] 8

9 CHAPTER 5d. PROBABILIT DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slde No. 6 Random Varable L w P Example: Smply Supported Beam The smply supported beam s subjected to the external loadng w and P as shown n the fgure. The probablstc characterstcs of the basc random varables are as follows: Mean 30 0 COV Standard Devaton Type Determnstc Normal Lognormal CHAPTER 5d. PROBABILIT DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slde No. 7 Example (cont d): Smply Supported Beam w P L/ wl PL M = + =.5w P 8 4 L 9

10 CHAPTER 5d. PROBABILIT DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slde No. 8 Example (cont d): Smply Supported Beam. Smulate the desgn moment M for 0 values.. Also, Fnd the mean, varance, standard devaton, and coeffcent of varaton of M usng the smulated sample values. wl PL M = + =.5w P 8 4 CHAPTER 5d. PROBABILIT DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slde No. 9 Example (cont d): Smply Supported Beam Mean (w ) = Mean (P ) = 0 Stdev (w ) = 0. Stdev (P ) = 3 u u w P M Mean (M ) = kp-ft M =.5w P Varance (M ) = 77.5 kp-ft Stdev (M ) = 7.0 kp- ft COV (M ) =

11 CHAPTER 5d. PROBABILIT DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slde No. 0 Example (cont d): Smply Supported Beam Sample Calculatons Consder the second row n the table: a) w: s normal u = F ( w ) = Φ ( z ) or Therefore, w = µ W W x µ z = σ + σ W z = = + 0. Φ = 0. Φ W w µ = Φ σw W W µ + σ Φ W W ( u ) ( ) = + 0. Φ ( ) ( ) = 0.(.39). 7 µ CHAPTER 5d. PROBABILIT DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slde No. = ln Sample Calculatons σ 3 B) P: s lognormal σ ln = ln + = µ = + 0 = ln(0) 0.49 =. ( µ ) σ ( ) 9846 ln P µ u = Φ or ln µ σ σ P = + Φ or [ µ + σ Φ ( u )] [ Φ ( ) P = e = e ] [ Φ ( ) ] [ ] = e = e =.957 ( u )

12 CHAPTER 5d. PROBABILIT DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slde No. Example (cont d): Smply Supported Beam Sample Calculatons Consder the the second row n the table C) M: M =.5w + 7.5P =.5.7 ( ) + 7.5(.957) = kp - ft CHAPTER 5d. PROBABILIT DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slde No. 3 Example (cont d): Smply Supported Beam Mean (w ) = Mean (P ) = 0 Stdev (w ) = 0. Stdev (P ) = 3 u u w P M Mean (M ) = kp-ft M =.5w P Varance (M ) = 77.5 kp-ft Stdev (M ) = 7.0 kp- ft COV (M ) = 0.07