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Probability and Limit States

Probability and Limit States. Introduction to Engineering Systems Lecture 6 (9/14/2009). Prof. Andrés Tovar. Announcements. should be (1.25, 27). Print out Learning Center documents, read them, and bring them to Learning Center There’s a typo in HW2. Note on polyfit. x = 0:.2:1;

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Probability and Limit States

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  1. Probability and Limit States Introduction to Engineering Systems Lecture 6 (9/14/2009) Prof. Andrés Tovar

  2. Announcements should be (1.25, 27) Print out Learning Center documents, read them, and bring them to Learning Center There’s a typo in HW2 Betting on a Design

  3. Note on polyfit x = 0:.2:1; y_exp = [1.4 1.1 3.3 2.4 3.8 3.0]; c = polyfit(x, y_exp, 1); m = c(1); % slope b = c(2); % intercept y_fit = m*x + b; plot(x, y_exp, 'o') % plot experimental data hold on plot(x, y_fit, 'r') % plot best-fit line hold off • Note that polyfit only finds coefficients of line (slope and intercept). It does not plot the line. • need to add plot command to do this Betting on a Design

  4. From last class • Basis function determines the empirical model. • Statistic is a numerical value that characterizes a dataset. • Mean or average (see also median and mode) • Standard deviation • Coefficient of variation • Uncertainty can be epistemic or aleatory. • Error can be systematic or random. Betting on a Design

  5. Firing at a Target • = 17.2 m • = 1.4 m Adjust the pullback distance X of the slingshot to hit a target at a distance D downrange. How likely is it that a softball launched with a pullback of 1 m will land within 1 m of 17 m? Betting on a Design

  6. Where to Aim the Slingshot? win $1 lose $5 • Suppose that you are required to take the following bet: • You will be paid $1 to launch a ball to less than 18 m. • But you will be fined $5 for each launch shorter than 16 m • What target distance should you aim for to maximize your gain? • How would you solve it? • What would you do differently if the fine were increased to $50? Betting on a Design

  7. Probability number of successful outcomes estimated experimental probability of winning = total number of trials = probability of losing 1 – probability of winning • The relative frequency of possible outcomes • It is measured on a scale from 0 to 1 • Estimate probability by running an experiment with multiple trials • and counting number of favorable/unfavorable outcomes Betting on a Design

  8. Coin toss • Suppose we run an experiment where we toss a coin 3 times H, H, T • What are the possible probabilities of the coin coming up “heads” that we could estimate from this experiment? 2/3 ≈ 0.6667 = 66.67% • What effect would using more trials have on our estimate? 1/2 = 0.5 = 50% Betting on a Design

  9. More games 1/36 http://www.roulettebonuscode.com/ http://craps.gamble-world.com/ Theoretical probability of single events Betting on a Design

  10. Example: Rolling a Die % simulation n = 100000; r = randi(6,1,n); wins = sum(r==1|r==6); losses = sum(r==2|r==3|r==4|r==5); simulated_gain = (5*wins - 2*losses)/n % theory p = 2; a = 5; q = 4; b = -2; theoretical_gain = (p*a + q*b)/(p + q) %#ok<*NOPTS> • Given a fair 6-sided die, what is your expected gain if: • you win $5 if you roll 1 or 6 • you lose $2 if you roll 2, 3, 4, or 5 • Should you take this bet? Betting on a Design

  11. Expected Gain (or Loss) From a Bet expected gain • Huygens (again), 1656 • Given • p chances of winning a sum of money a • q chances of winning a sum of money b Betting on a Design

  12. Estimating Probabilities for the Slingshot • For firing a slingshot with a pullback of 1 m. Estimate the probability that: • ball lands within 1 m of the target at 17 m. • ball lands short of this range (>18 m) • ball lands beyond this range (<16 m) Betting on a Design

  13. Bin Counts Betting on a Design

  14. Histogram Betting on a Design

  15. Probabilities Betting on a Design

  16. Theory of the Bell Curve: Normal Distribution flips from concave up to concave down 0.1% 2.1% 2.1% 0.1% 13.6% 13.6% 99.7% of distribution is between  3 99.9% of distribution is less than + 3 Betting on a Design

  17. Experimental vs. theoretical distribution % distances of the softball D = [17.5 19.0 16.4 19.3 16.6 16.0 17.4 16.7 18.1 17.5 ... 15.1 14.2 17.4 15.7 17.8 19.3 18.5 15.7 17.9 17.0]; m = mean(D); % mean s = std(D); % standard deviation % experimental probability density y = 14:1:20; n = histc(D,y); bar(y+1/2,n/length(D),1); hold on xlabel('distanced (m)') ylabel('probabilityp(d)') % normal probability density itv = 0.0001; x = m-3*s:itv:m+3*s; p = 1/(s*sqrt(2*pi))*exp(-(x-m).^2/(2*s^2)); plot(x,p,'r.'); hold off p_theo=sum(p(x>=16 & x<=18))*itv p(16≤d≤18) = 0.55 (experimental) p(16≤d≤18) = 0.53 (normal distribution) Betting on a Design

  18. The probabilities % distances of the softball D = [17.5 19.0 16.4 19.3 16.6 16.0 17.4 16.7 18.1 17.5 ... 15.1 14.2 17.4 15.7 17.8 19.3 18.5 15.7 17.9 17.0]; m = mean(D); % mean s = std(D); % standard deviation % plots itv = 0.0001; x = (m-3*s:itv:m+3*s); px = pdf('norm',m-3*s:itv:m+3*s,m,s); % probability density function figure(1); plot(x,px,'r.') % cumulative distribution function pxc = cdf('norm',m-3*s:itv:m+3*s,m,s); figure(2); plot(x,pxc,'b.') % results px16 = cdf('norm',16,m,s) % x<=16 pxm = cdf('norm',16:18,m,s); px17 = pxm(3)-pxm(1) % 16<x<=18 px18 = 1-cdf('norm',18,m,s) % x>18 Betting on a Design

  19. Should we take the bet? p= 0.5280, a = 1 q = 0.2016, b = -5 Betting on a Design

  20. DESIGN STAGE CONSTRUCTION & VERIFICATION Investigate Designs using Model Construct Design Optimize Design Experimentally Verify Behavior Predict Behavior Where we are in the tower design process MODEL DEVELOPMENT Gather Data Develop Model Verify Model Betting on a Design

  21. Tower Bracing Betting on a Design

  22. Design Considerations Betting on a Design

  23. Limit States • Limit state: objective on the performance or behavior of a design • The consequences of failing to meet limit states can be very high • customer dissatisfaction (loss of business) • product recall • loss of property • loss of life • Terminology (“limit state design”) comes from structural engineering, but the concept cuts across all fields Betting on a Design

  24. Limit States for Skyscraper Design Betting on a Design

  25. Distribution of Tower Stiffness 0.07 N/mm Betting on a Design

  26. Histogram of Tower Trial Stiffnesses m = 0.07 N/mm s = 0.02 N/mm Betting on a Design

  27. How Good of a Fit for Tower Stiffnesses?  = 0.07  = 0.02 • Pretty good fit! • We can use the normal distribution model to estimate probabilities Betting on a Design

  28. Where to Aim for Tower Stiffness expensive pass limit state = 4.5N/19mm bracing cost fail cheap Betting on a Design

  29. Confidence in Meeting Limit State Use probabilities from normal distribution to determine where to aim your design Betting on a Design

  30. This week • The tower design problem • Factor of safety • Constraints • Efficiency • E = S/B • S = kbr/kubr: stiffness ratio • B = Lbr/Lubr: bracing ratio • Introduction to design optimization Betting on a Design

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