CARLO SIMULATION 1 ENCE

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1 CHAPTER Duxbury Thomson Learnng Makng Hard Decson MONTE CARLO SMULATON Thrd Edton A. J. Clark School of Engneerng Department of Cvl and Envronmental Engneerng b FALL 00 By Dr. brahm. Assakkaf ENCE 67 Decson Analyss for Engneerng Department of Cvl and Envronmental Engneerng Unversty of Maryland, College Park CHAPTER b. MONTE CARLO SMULATON Slde No. ENCE 67 Assakkaf

2 CHAPTER b. MONTE CARLO SMULATON Slde No. Methodology of Modelng Uncertanty The Methodology of Modelng Uncertanty s descrbed n fve chapters that manly concentratng on how to model uncertanty usng probabltes and nformaton as follows: ENCE 67 Assakkaf Detaled Steps Probablty Bascs: revews fundamental probablty concepts. Subjectve probablty: translates belefs & feelngs about uncertanty n probablty for use n decson modelng. Theoretcal Probablty Models: helps wth representng uncertanty n decson modelng Usng Data: uses hstorcal data for developng probablty dstrbutons Monte Carlo Smulaton: to gve the decson-maker a far dea about the probabltes assocated wth varous outcomes. Value of nformaton: explores the value of nformaton wthn the decson-analyss framework. Chapter 7 Chapter 8 Chapter 9 Chapter 0 Chapter Chapter CHAPTER b. MONTE CARLO SMULATON Slde No. ntroducton to Smulaton ENCE 67 Assakkaf Smulaton Technques Smulaton s a process of replcatng the real world based on a set of assumptons and conceved models of realty. Smulaton can be performed ether: Expermentally, or Theoretcally n practce, theoretcal smulaton s performed (nexpensve).

3 CHAPTER b. MONTE CARLO SMULATON Slde No. 4 ntroducton to Smulaton ENCE 67 Assakkaf Smulaton may be appled n engneerng to predct or study the performance and response of a system. Smulaton can be used to verfy the accuracy of structural relablty methods wth lttle background n probablty and statstcs. A smulaton method can provde estmates for any problem, whereas analytcal methods may not always converge n ther teratons CHAPTER b. MONTE CARLO SMULATON Slde No. 5 ntroducton to Smulaton ENCE 67 Assakkaf Need for Smulaton and Data Analyss Examples: Transportaton engneers frequently use traffc counts at ntersecton or accdent data for varous confguratons of control sgnals n desgnng hghways. However, f these data are nsuffcent or costly, they resort to smulaton. Envronmental engneers collect data on water qualty and analyze these data to decde upon the type of water treatment that s needed. Unfortunately, stream-flow records often do not nclude extreme floods that are mportant n evaluatng flood rsk. For ths reason, they use smulated data to help makng decsons.

4 CHAPTER b. MONTE CARLO SMULATON Slde No. 6 ntroducton to Smulaton ENCE 67 Assakkaf Example: Smulaton by Con Flppng Water qualty for a partcular locaton on a rver A acceptable U unacceptable Assumptons About 50% of the tme acceptable 50% of the tme unacceptable Data was obtaned for the last two weeks (4 days)» AAUAUUAUUAAAUA Damage to aquatc lfe occurs f the water qualty s unacceptable for three or more consecutve days CHAPTER b. MONTE CARLO SMULATON Slde No. 7 ntroducton to Smulaton ENCE 67 Assakkaf Concluson Accordng to the real data, the water qualty was of acceptable qualty 8/ (57%) of the days and there were no nstances of aquatc damage. Should the engneer, therefore, beleve that aquatc damage wll not occur n the future? Of course not!

5 CHAPTER b. MONTE CARLO SMULATON Slde No. 8 ntroducton to Smulaton ENCE 67 Assakkaf Use of smulaton to make a decson based on the probablty of aquatc damage n the future Flppng a con 56 tmes produces the 8- week sequence: H head (acceptable) T tal (unacceptable) HHTHTHHHHHTHTHTTHTTTHTHHHTHT THTTTTHHTHHHTHTHTHTTHHHTTHTH CHAPTER b. MONTE CARLO SMULATON Slde No. 9 ntroducton to Smulaton ENCE 67 Assakkaf Conclusons f a tal s consdered an acceptable water qualty, then two occurrences of unacceptable qualty happened n 8 weeks Ths represents 4 weeks or tmes a year

6 CHAPTER b. MONTE CARLO SMULATON Slde No. 0 Generaton of Random Numbers ENCE 67 Assakkaf Flp of a Con For one event, only one of the two possble outcomes can occur, H or T Probablty H Outcome T CHAPTER b. MONTE CARLO SMULATON Slde No. Generaton of Random Numbers ENCE 67 Assakkaf Flppng of Two Cons Outcome (T,T), (H,T) or (T,H), and (H,H) Let (T,T) 0 (H,T) or (T,H) (H,H) number of heads Then, the probablty P(x) can be graphed as

7 CHAPTER b. MONTE CARLO SMULATON Slde No. Generaton of Random Numbers ENCE 67 Assakkaf Probablty Outcome, CHAPTER b. MONTE CARLO SMULATON Slde No. Generaton of Random Numbers ENCE 67 Assakkaf Rollng of Dce

8 CHAPTER b. MONTE CARLO SMULATON Slde No. 4 Generaton of Random Numbers ENCE 67 Assakkaf Rollng of a Sngle De f the random events were generated wth the roll of sngle de, one of the sx outcomes s possble. f the de s far, each outcome y s equally lkely, and each would have a probablty of /6. Graphcally, ths can be presented as CHAPTER b. MONTE CARLO SMULATON Slde No. 5 Generaton of Random Numbers ENCE 67 Assakkaf Probablty Outcome,

9 CHAPTER b. MONTE CARLO SMULATON Slde No. 6 Generaton of Random Numbers ENCE 67 Assakkaf Rollng of Two Dce CHAPTER b. MONTE CARLO SMULATON Slde No. 7 Generaton of Random Numbers ENCE 67 Assakkaf Rollng of Two Dce (cont d) f a par of dce rolled smultaneously, the probablty of the sum of dots from the two dce would appear graphcally as shown n the followng fgure:

10 CHAPTER b. MONTE CARLO SMULATON Slde No. 8 Generaton of Random Numbers ENCE 67 Assakkaf Outcome Probablty /6 /6 /6 4/6 5/6 6/6 5/6 4/6 /6 /6 /6 Probablty Outcome, Z CHAPTER b. MONTE CARLO SMULATON Slde No. 9 Generaton of Random Numbers ENCE 67 Assakkaf Dscrete versus Contnuous Values n each of the prevous cases, only ntegers values were possble. For example, f a sngle de s rolled, a value of 4.6 s not possble. Values from a flp of con or a roll of a de are dscrete (.e., ntegers,, ). Examples: Engneerng Cases: Number of traffc fataltes Number of floods per decade Number of earthquakes above 6 per century

11 CHAPTER b. MONTE CARLO SMULATON Slde No. 0 Generaton of Random Numbers ENCE 67 Assakkaf Dscrete versus Contnuous Values Values on contnuum could be generated wth spnner (some board games) placed over a 600 protractor. Examples: Engneerng Cases: Stoppng dstance of as car Magntude of a flood Compresson strength of concrete Weght of fertlzer used per acre CHAPTER b. MONTE CARLO SMULATON Slde No. Generaton of Random Numbers ENCE 67 Assakkaf Probablty of the outcome of a spn s / Probablty P (A ) Outcome, A

12 CHAPTER b. MONTE CARLO SMULATON Slde No. Generaton of Random Numbers ENCE 67 Assakkaf Transformaton of the angle A to a new varable B can be accomplshed by requrng that B takes values from 0 to. Ths can be done usng B A/ Probablty Outcome, B CHAPTER b. MONTE CARLO SMULATON Slde No. Computer Generaton of Random Numbers ENCE 67 Assakkaf Computer software packages are avalable The generated random numbers from these packages are called pseudo random numbers These numbers are generated from a welldefned and predctable process

13 CHAPTER b. MONTE CARLO SMULATON Slde No. 4 Computer Generaton of Random Numbers Mdsquare Method ENCE 67 Assakkaf 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 Use the four dgts n the mddle as the new random number. Repeat steps and to generate as many numbers as necessary CHAPTER b. MONTE CARLO SMULATON Slde No. 5 Computer Generaton of Random Numbers Example : Mdsquare Method Consder the seed number 89. Ths value would produce the followng: ENCE 67 Assakkaf

14 CHAPTER b. MONTE CARLO SMULATON Slde No. 6 Computer Generaton of Random Numbers Example : Mdsquare Method ENCE 67 Assakkaf Consder the seed number 500. Ths value would produce the followng: The above random-number sequence s not good for statstcal purposes. Other more relable methods for generatng random numbers are avalable. CHAPTER b. MONTE CARLO SMULATON Slde No. 7 Transformaton of Random Varables How can flps of a con be generated wth a de? ENCE 67 Assakkaf Ths can be accomplshed by transformng the value of the de to the value of the con. The de s rolled and an occurrence of a,, or would consttute a head, whle 4, 5, or 6 would consttute a tal. Ths can be presented graphcally as

15 CHAPTER b. MONTE CARLO SMULATON Slde No. 8 Transformaton of Random Varables 7 ENCE 67 Assakkaf 6 5 De Outcome 4 0 H Con Outcome T CHAPTER b. MONTE CARLO SMULATON Slde No. 9 Transformaton of Random Varables 40 ENCE 67 Assakkaf Outcome, A Outcome, B

16 CHAPTER b. MONTE CARLO SMULATON Slde No. 0 Smulaton and Probablty Dstrbuton Need for Smulaton ENCE 67 Assakkaf 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 b. MONTE CARLO SMULATON Slde No. Smulaton and Probablty Dstrbuton Monte Carlo Smulaton Monte Carlo smulaton has sx essental elements: ENCE 67 Assakkaf. Defnng the problem n terms of all the random varables,. Quantfyng the probablstc characterstcs of all the random varables (.e., mean, COV, dstrbuton type),. Generatng the values of these random varables

17 CHAPTER b. MONTE CARLO SMULATON Slde No. Smulaton and Probablty Dstrbuton Monte Carlo Smulaton (cont d) ENCE 67 Assakkaf 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 b. MONTE CARLO SMULATON Slde No. Smulaton and Probablty Dstrbuton Formulaton of the Problem Consder a smply supported beam as shown P w ENCE 67 Assakkaf L L/

18 CHAPTER b. MONTE CARLO SMULATON Slde No. 4 Smulaton and Probablty Dstrbuton Assume both w and P are random varables. ENCE 67 Assakkaf 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 b. MONTE CARLO SMULATON Slde No. 5 Smulaton and Probablty Dstrbuton f the span of the beam s 0 feet, the expresson for the desgn moment can wrtten as ENCE 67 Assakkaf M wl PL w + 7.5P W and P n ths case are called basc random varables.

19 CHAPTER b. MONTE CARLO SMULATON Slde No. 6 Smulaton and Probablty Dstrbuton Generaton of Random Varables ENCE 67 Assakkaf 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 b. MONTE CARLO SMULATON Slde No. 7 Smulaton and Probablty Dstrbuton ENCE 67 Assakkaf 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. Use the four dgts n the mddle as the new random number. 4. Repeat steps and to generate as many numbers as necessary

20 CHAPTER b. MONTE CARLO SMULATON Slde No. 8 Smulaton and Probablty Dstrbuton Example : Mdsquare Method Consder the seed number 89. Ths value would produce the followng: ENCE 67 Assakkaf CHAPTER b. MONTE CARLO SMULATON Slde No. 9 Smulaton and Probablty Dstrbuton Transformaton of Unform Random Numbers The Unform Dstrbuton 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 a + b and σ ( b a) ENCE 67 Assakkaf

21 CHAPTER b. MONTE CARLO SMULATON Slde No. 40 Smulaton and Probablty Dstrbuton nverse Transformaton Technque or nverse CDF Method ENCE 67 Assakkaf n 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 b. MONTE CARLO SMULATON Slde No. 4 ENCE 67 Assakkaf F U (u ) F (x ) U f U (u ) f (x ) x F ( u ) U u 0 x

22 CHAPTER b. MONTE CARLO SMULATON Slde No. 4 Smulaton and Probablty Dstrbuton Example: Normal Dstrbuton ENCE 67 Assakkaf f s normally dstrbuted, that s ~ N(,σ), then Z (- )/σ s a standard normal varate, that s, Z ~ N(0,). t can be shown that u F or Thus, ( x ) Φ( z ) x x z σ + σ x Φ σ z + σ Φ ( u ) CHAPTER b. MONTE CARLO SMULATON Slde No. 4 Smulaton and Probablty Dstrbuton Example: Lognormal Dstrbuton ENCE 67 Assakkaf f s lognormally dstrbuted, that s ~ LN(,σ ), then Z (ln- )/σ s a standard normal varate, that s, Z ~ N(0,). t can be shown that ln x u F ( x ) Φ( z ) Φ σ or Thus, ln ( x ) + σ Φ ( u ) x e [ + σ Φ ( u )]

23 CHAPTER b. MONTE CARLO SMULATON Slde No. 44 Smulaton and Probablty Dstrbuton Example: Smply Supported Beam ENCE 67 Assakkaf 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: Random Varable L Mean 0 COV - Standard Devaton - Dstrbuton Type Determnstc w Normal P Lognormal CHAPTER b. MONTE CARLO SMULATON Slde No. 45 Smulaton and Probablty Dstrbuton ENCE 67 Assakkaf Example (cont d): Smply Supported Beam w P L/ wl PL M +.5w P 8 4 L

24 CHAPTER b. MONTE CARLO SMULATON Slde No. 46 Smulaton and Probablty Dstrbuton Example (cont d): Smply Supported Beam ENCE 67 Assakkaf. 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 b. MONTE CARLO SMULATON Slde No. 47 Smulaton and Probablty Dstrbuton Example (cont d): Smply Supported Beam Mean (w ) Mean (P ) 0 Stdev (w ) 0. Stdev (P ) u u w P M Mean (M ) 80.9 kp-ft M.5w P Varance (M ) 77.5 kp-ft Stdev (M ) 7.0 kp- ft COV (M ) 0.07 ENCE 67 Assakkaf

25 CHAPTER b. MONTE CARLO SMULATON Slde No. 48 Smulaton and Probablty Dstrbuton 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 W W ( ) ( ) x z σ + σ W z + 0. Φ 0. Φ W w Φ σw W W ENCE 67 Assakkaf + σ Φ W W ( u ) ( ) + 0. Φ ( ) ( ) 0.(.9). 7 CHAPTER b. MONTE CARLO SMULATON Slde No. 49 Smulaton and Probablty Dstrbuton ln Sample Calculatons B) P: s lognormal σ ln(0) ( ) σ ( ) 9846 σ ln + ENCE 67 Assakkaf ln ln P u Φ or ln σ σ P + Φ or [ + σ Φ ( u )] [ Φ ( ) P e e ] [ Φ ( ) ] [ ] e e.957 ( u )

26 CHAPTER b. MONTE CARLO SMULATON Slde No. 50 Smulaton and Probablty Dstrbuton 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 ENCE 67 Assakkaf CHAPTER b. MONTE CARLO SMULATON Slde No. 5 Functons of Random Varables Approxmate Methods Taylor Seres Expanson A Taylor seres s commonly used n engneerng analyss to approxmate functons that do not have closed form soluton. The Taylor seres s gven by f h! () ( ( ) ( ) ( ) ) ( ( ) ) ( n x ( ) ) 0 + h f x0 + hf x0 + f x0 + f x f ( x0 ) + Rn+ where x 0 base value or startng value x the pont at whch the value of the functon s needed h x x 0 dstance between x 0 and x (step sze) n! factoral of n n(n-) (n ) n h n! f (n) ndcates the n th dervatve of the functon f(x) R n+ the remander of Taylor seres expanson h! ENCE 67 Assakkaf

27 CHAPTER b. MONTE CARLO SMULATON Slde No. 5 Functons of Random Varables Approxmate Methods Taylor Seres Expanson Frst-order approxmaton ENCE 67 Assakkaf ( ( x + h) f ( x ) hf ) ( ) f x0 Second-order approxmaton h f ! () ( ( x + h) f ( x ) + hf ( x ) f ) ( x ) Thrd-order approxmaton h h f x0 + h f x0 + hf x0 + f x0 +!! () ( ( ) ( ) ( ) ) ( ( ) f ) ( x ) 0 0 CHAPTER b. MONTE CARLO SMULATON Slde No. 5 Functons of Random Varables ENCE 67 Assakkaf Approxmate Methods The Taylor seres expanson can be used to approxmate the mean and varance of a functon of random varables g() Two cases to be consdered:. Sngle random varable. Multple random varables, a random vector

28 CHAPTER b. MONTE CARLO SMULATON Slde No. 54 Functons of Random Varables ENCE 67 Assakkaf or g g Approxmate Methods Sngle Random Varable ( ) g( + h) g( ) ( ) + [ ] The Taylor seres expanson of a functon g() about the mean of (E()) s gven by ( ) dg d dg + h d + k ( ) d g( ) d g( ) [ ] + h! d g d ( ) d k! h k! [ ] k k d k k d g d ( ) k CHAPTER b. MONTE CARLO SMULATON Slde No. 55 Functons of Random Varables ENCE 67 Assakkaf Approxmate Methods Sngle Random Varable g( ) g [ E( )] + [ E( )] k! g( ) g [ E( )] ( ) dg d [ E( )] + [ E( )] k k d g d E( ) ( ) k E( ) f the seres s truncated at the second term, then ( ) dg d + [ E( )] E ( ) d g d ( ) E( )

29 CHAPTER b. MONTE CARLO SMULATON Slde No. 56 Functons of Random Varables ENCE 67 Assakkaf Approxmate Methods Sngle Random Varable g( ) g Hence, E E [ E( )] + [ E( )] ( ) dg d E( ) Takng the expectaton of both sdes, and notng that [ E( )] E( ) E[ E( )] E( ) E( ) ( ) E[ g( )] g[ E( )] g( ) 0 CHAPTER b. MONTE CARLO SMULATON Slde No. 57 Functons of Random Varables ENCE 67 Assakkaf Approxmate Methods Sngle Random Varable dg [ ( )] [ ( )] ( ) ( ) g E + E g Hence, E Var [ g[ E( ) ] Var[ g( )] ( ) Var [ E( )] ( ) dg d d 0 E( ) E( ) Takng the varances of both sdes, and notng that ( ) dg d E( ) Var ( )

30 CHAPTER b. MONTE CARLO SMULATON Slde No. 58 Functons of Random Varables ENCE 67 Assakkaf Approxmate Methods (sngle RV) Frst-order (approxmate ) Mean ( ) g[ E( )] E Frst-order (approxmate) Varance Var ( ) ( ) dg σ Var d E( ) ( ) CHAPTER b. MONTE CARLO SMULATON Slde No. 59 Functons of Random Varables ENCE 67 Assakkaf Example: Pressure of Ocean Waves The maxmum mpact pressure of ocean waves on coastal structures may be determned by ρkv ρ max.7 D Where ρ densty of water, K length of hypothetcal pston, D thckness of ar cushon, V horzontal velocty of advancng wave. Suppose that the mean crest velocty V s 4.5 ft/sec wth COV of 0.. ρ, K, and D are constants. f ρ.96 slugs/cu ft, and the rato K/D 5, determne the mean and standard devaton of the peak mpact pressure.

31 CHAPTER b. MONTE CARLO SMULATON Slde No. 60 Functons of Random Varables ENCE 67 Assakkaf E Example (cont d): Pressure of Ocean Waves ( ρ ) g[ E( V )].7(.96)( 5)( 4.5) and dρ dv Var σ max max ρ max V 4.5 ρkv.7 D dρ dv (.7) (.7)(.96)( 5)( 4.5) max ( ρ ) Var( V ) (,666.98) ( ) max ρmax d dv V 4.5,50,846.,500. psf ρkv D psf,666.98,50,846. psf CHAPTER b. MONTE CARLO SMULATON Slde No. 6 Functons of Random Varables ENCE 67 Assakkaf Approxmate Methods (Random Vector) Frst-order (approxmate ) Mean ( ) g[ E( ), E( ),..., E( )] E n Var Frst-order (approxmate) Varance ( ) σ n n g ( ) j E ( ) g ( ) j E ( ) Cov (, ) j

32 CHAPTER b. MONTE CARLO SMULATON Slde No. 6 Functons of Random Varables ENCE 67 Assakkaf Approxmate Methods (Random Vector) Frst-order (approxmate) Varance f the s are uncorrelated (statstcally ndependent), then Var ( ) σ n g ( ) E( ) Var ( ) CHAPTER b. MONTE CARLO SMULATON Slde No. 6 Functons of Random Varables ENCE 67 Assakkaf Example : Assume that the random varable can be represented by the followng relatonshp: / where,, and are statstcally ndependent random varables wth mean values of.0,.5, and 0.8, respectvely, and correspondng standard devatons of 0., 0., and 0.5, respectvely. Fnd the frst-order mean and standard devaton of.

33 CHAPTER b. MONTE CARLO SMULATON Slde No. 64 ENCE 67 Assakkaf Example (cont d): ( ) ( ) ( ) ( ) [ ] ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) / / / / / / / / / / ,E E, E E g / Functons of Random Varables CHAPTER b. MONTE CARLO SMULATON Slde No. 65 ENCE 67 Assakkaf Example (cont d): ( ) ( ) [ ] ( ) ()( )( ) [ ] ()( ) ( ) (.5) / / / / / / Functons of Random Varables

34 CHAPTER b. MONTE CARLO SMULATON Slde No. 66 Functons of Random Varables ENCE 67 Assakkaf Var σ ( ) Example (cont d): Var Var Var ( ) σ ( 0.) 0.0 ( ) σ ( 0.) 0.04 ( ) σ ( 0.5) σ g g ( ) ( ) E( ) E( ) ( ) ( ) ( ) E( ) ( ) ( 4.67)( 0.0) ( 0.04) ( 0.05) Var Var g + Var ( ) g E( ) Var ( ) CHAPTER b. MONTE CARLO SMULATON Slde No. 67 Functons of Random Varables ENCE 67 Assakkaf Example : The stress F n a beam subjected to an external bendng moment M s My F where y s the dstance from the neutral axs of the cross secton of the beam to the pont where the stress s calculated, and s the centrodal moment of nerta of the cross secton. Assume that M and are random varables wth means M and, respectvely, and varances σ M and σ, respectvely.

35 CHAPTER b. MONTE CARLO SMULATON Slde No. 68 ENCE 67 Assakkaf Example (cont d): Determne the mean and varance of F based on frst-order approxmaton. M M F y My My F y y M My M F y Functons of Random Varables CHAPTER b. MONTE CARLO SMULATON Slde No. 69 ENCE 67 Assakkaf Example (cont d): ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) E E E σ σ Var Var Var σ Var M M F y y F g M M F g g F + + Functons of Random Varables

36 CHAPTER b. MONTE CARLO SMULATON Slde No. 70 Multvarable Smulaton ENCE 67 Assakkaf Smulaton can be used to study the probablstc characterstcs of a functon of random varables. t can provdes nformaton about the dstrbutons of random varables that s beyond the ablty of theory. Theoretcal relatonshps are often based on restrctve assumptons, such as normal dstrbutons, that may not be vald for a gven problem. CHAPTER b. MONTE CARLO SMULATON Slde No. 7 Multvarable Smulaton ENCE 67 Assakkaf Stress at Extreme Fbers of a Beam The stress at the extreme fbers of steel beam s gven by Mc σ To estmate the mean and standard devaton of σ, we can use the frst-order approxmaton as dscussed prevously, regardless of the dstrbuton types of the basc random varables c and.

37 CHAPTER b. MONTE CARLO SMULATON Slde No. 7 Multvarable Smulaton ENCE 67 Assakkaf Stress at Extreme Fbers of a Beam Smulaton can also be used to study the probablstc characterstcs of σ, such as the mean and standard devaton. However, the dstrbuton types of the basc random varables c and are requred. CHAPTER b. MONTE CARLO SMULATON Slde No. 7 Multvarable Smulaton ENCE 67 Assakkaf Stress at Extreme Fbers of a Beam Random Varable Mean Standard Devaton Dstrbuton Type c Normal M Lognormal Normal

38 CHAPTER b. MONTE CARLO SMULATON Slde No. 74 Multvarable Smulaton ENCE 67 Assakkaf Stress at Extreme Fbers of a Beam Frst-order Approxmate mean and standard devaton of M Mc 000( 0) σ σ M σ c σ Var c M , 000, 000 Mc 000( 0) σ 0.0, ( 000) M ( σ) ( 0.000)( 900) + 9( 0.5) ( 80) Standard Devaton (σ ) σ c σ M CHAPTER b. MONTE CARLO SMULATON Slde No. 75 Multvarable Smulaton ENCE 67 Assakkaf Stress at Extreme Fbers of a Beam Smulaton result for the mean and standard devaton of M For 000 smulaton cycles, Mean of σ 0.5 Standard Devaton of σ 9.67 # Cycles u u u M c σ

39 CHAPTER b. MONTE CARLO SMULATON Slde No. 76 Multvarable Smulaton ENCE 67 Assakkaf Stress at Extreme Fbers of a Beam Comparson Between Approxmate Method and Smulaton Mean of σ Mc σ Standard Devaton of σ Approxmaton Smulaton