EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Probability and Limit States Introduction to Engineering Systems Lecture 6 (9/14/2009) Prof. Andrés Tovar

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Announcements Print out Learning Center documents, read them, and bring them to Learning Center There’s a typo in HW2 Betting on a Design2 should be (1.25, 27)

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Note on polyfit 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 Design3 x = 0:.2:1; y_exp = [ ]; 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

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame 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 Design4

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Firing at a Target 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 Design5  = 17.2 m  = 1.4 m

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Where to Aim the Slingshot? 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 win $1 lose $5 6

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Probability 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 estimated experimental probability of winning number of successful outcomes total number of trials = probability of losing1 – probability of winning = 7

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame 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 ≈ = 66.67% What effect would using more trials have on our estimate? 1/2 = 0.5 = 50% Betting on a Design8

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame More games Betting on a Design9 1/36 Theoretical probability of single events

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Example: Rolling a Die 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 Design10 % simulation n = ; 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

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Expected Gain (or Loss) From a Bet 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 expected gain 11

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame 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 Design12

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Bin Counts Betting on a Design13

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Histogram Betting on a Design14

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Probabilities Betting on a Design15

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Theory of the Bell Curve: Normal Distribution Betting on a Design16 flips from concave up to concave down 2.1% 13.6% 2.1%0.1% 99.7% of distribution is between  3  99.9% of distribution is less than  + 3 

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Experimental vs. theoretical distribution Betting on a Design17 % distances of the softball D = [ ]; 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('distance d (m)') ylabel('probability p(d)') % normal probability density itv = ; 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)

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame The probabilities Betting on a Design18 % distances of the softball D = [ ]; m = mean(D); % mean s = std(D); % standard deviation % plots itv = ; 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

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Should we take the bet? Betting on a Design19 p = , a = 1 q = , b = -5

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Where we are in the tower design process Betting on a Design20 Gather Data Develop Model Verify Model MODEL DEVELOPMENT Investigate Designs using Model Optimize Design Predict Behavior DESIGN STAGE Construct Design Experimentally Verify Behavior CONSTRUCTION & VERIFICATION

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Tower Bracing Betting on a Design21

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Design Considerations Betting on a Design22

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame 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 Design23

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Limit States for Skyscraper Design Betting on a Design24

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Distribution of Tower Stiffness Betting on a Design N/mm

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Histogram of Tower Trial Stiffnesses Betting on a Design26  = 0.07 N/mm  = 0.02 N/mm

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame How Good of a Fit for Tower Stiffnesses? Pretty good fit! –We can use the normal distribution model to estimate probabilities Betting on a Design27  = 0.07  = 0.02

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Where to Aim for Tower Stiffness Betting on a Design28 fail pass limit state = 4.5N/19mm bracing cost expensive cheap

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Confidence in Meeting Limit State Use probabilities from normal distribution to determine where to aim your design 29 Confidence in exceeding limit state Probability of Failure Aim design at 50.0% limit state value 84.2%15.8% limit state value +  97.8%2.2% limit state value + 2  99.9%0.1% limit state value + 3  Betting on a Design

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame This week The tower design problem –Factor of safety –Constraints –Efficiency E = S/B S = k br /k ubr : stiffness ratio B = L br /L ubr : bracing ratio Introduction to design optimization Betting on a Design30