1. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Design optimization: metrics and tower bracing Introduction to Engineering Systems Lecture 7 (9/18/2009) Prof. Andrés Tovar

2. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Announcements Deliverable and HW3 are due next week. HW4 will be posted Monday on Concourse. Read the project document posted on Concourse. Did you see all Matlab videos? –Check videos 11 and 12 –Check chapter 10 (statistics) Betting on a Design2

3. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Where’s the Bug? Betting on a Design 3 a = 2; b = 3; c = 5; r1 = (-b + sqrt(b^2 - 4*a*c))/2*a r2 = (-b - sqrt(b^2 - 4*a*c))/2*a

4. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Where’s the Bug? Betting on a Design 4 a = 2; b = 3; c = 5; r1 = (-b + sqrt(b^2 - 4*a*c))/2*a r2 = (-b - sqrt(b^2 - 4*a*c))/2*a MATLAB M-file names must be one word

5. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Where’s the Bug? Betting on a Design 5 x = 1:10; a = 2; b = 3; c = 5; y = a*x^2 + b*x + c

6. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Where’s the Bug? Betting on a Design 6 x = 1:10; a = 2; b = 3; c = 5; y = a*x.^2 + b*x + c need the “dot” for element-wise exponentiation (or multiplication, or division)

7. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Where’s the Bug? Betting on a Design 7 mean = sum(x)/length(x) c_o_v = std(x)/mean(x) “mean” is a predefined MATLAB function name. Don’t use it as a variable name!!! Note that “cov” is also a MATLAB function name. (Calculates something called a “covariance”)

8. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Testing for and Counting Values in Range Betting on a Design 8 >> x = [1 3 5 7 9] x = 1 3 5 7 9 >> x < 6 ans = 1 1 1 0 0 >> (x = 3) % "&" means "and" ans = 0 1 1 0 0 >> sum((x = 3)) % use sum to count ans = 2 see Ch 10 and Video 11

9. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Histograms Use arguments to hist to control number of “bins” See also histc to get bin counts Betting on a Design 9 d = [1.0 1.1 1.2... 2.1 2.2 2.3 2.4 2.5 2.6... 3.0 3.1]; hist(d)hist(d,3) hist(d,0.5:1:4.5) see Ch 10 and Video 12

10. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame From last class Experimental probability of an event is the ratio of the number of times the event occurs to the total number of trials, e.g., coin toss P(H)=2/3 Theoretical probability of an event is the ratio of the number of ways the event can occur to the total number of outcomes, e.g., coin toss P(H)=1/2 Betting on a Design10 P(16≤d≤18) = 0.55 (experimental) P(16≤d≤18) = 0.53 (theoretical) Normal distribution (PDF)

11. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame From last class Expected gain (G) based on probability (Huygens’ formula) Betting on a Design11 p = 0.5280, a = 1 q = 0.2016, b = -5

12. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Probability of failure Betting on a Design12 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  

13. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Design considerations Betting on a Design13 Limit state: (Civil Structural Engineering) objective on the performance or behavior of a design, e.g., Survivability, Serviceability, Habitability. Efficiency E = S/B S = k br /k ubr : stiffness ratio B = L br /L ubr : bracing ratio

14. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Cost Economic factors that influence design –Materials (both used or wasted) –Time: development/design and production/construction –Labor: human or automated, skilled vs. unskilled –Cost/complexity of construction, manufacturing process –Power –Disposal of waste or byproducts Betting on a Design14

15. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Constraints Additional design considerations –Size or dimensions (portability, transport, user dimensions, architecture, function) –Power source/consumption (in construction/production or operation/usage) –Time (production/construction or design time) –Materials (local availability) –Additional industry-specific standards Betting on a Design15 Pet architecture, Nayoga, Japan

16. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Tower Metrics: Bracing Ratio Factor by which bracing increases tower cost. Ratio of length of braces added L br to the design divided by the total length of vertical members L ubr in the design where L ubr = 4 columns/floor x N floors x 150 mm/column. Betting on a Design16

17. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Tower Metrics: Stiffness Ratio Factor by which bracing improves tower stiffness Betting on a Design17

18. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Efficiency Improvement in stiffness relative to cost. Higher efficiency leads to a better design. Betting on a Design18

19. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Bracing Scheme Options Design 1Design 2Design 3Design 4Design 5Design 6 S 458.35.6903.95.2903.91355.9 E 648.25.9639.15.2374.4517.3 19 Design 1 Design 6 (3-X braces) Design 2 Design 4 Design 3 Design 5

20. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Optimization of Tower Design 20 Need to First Understand: 1.How bracing affects our displacement and why? 2.How do different bracing schemes affect efficiency? 3.How does bracing placement affect our displacement & efficiency?

21. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Load Paths of Unbraced Structure 21 Shearing

22. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Load Paths of our Braced Structure Minimizes Shear Minimizes Deflections What types of bracing do you think will decrease the stiffness the most? 22

23. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Tower Behavior Under Specified Loading Does the tower represent the Axial or Shear Action Behavior? How can we use this information? 23

24. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Optimization Tips… 1.Bracing designs that reduce shear forces in the columns will be most successful 2.Efficiency over Stiffness  Form comparable stiffness ratios, chose option with higher Efficiency 24

25. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame How Does Bracing Affect Drift? Displacement is different at different floors Drift = deflection –Interstory: deflections at each story due to shearing –Cumulative: total deflections at each floor 25

26. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Bracing Location 26 Unbraced Braced @ Top Floor Braced @ 3rd Floor

27. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame What about Location of Bracing and Drift? Unbraced Top Floor Brace 3 rd Story Brace 5 th Floor 101.1 mm (19.2 mm) 77.3 mm (1.1 mm) 67.1 mm (18.5 mm) 4 th Floor 81.9 mm (21.2 mm) 76.2 mm (16.3 mm) 48.6 mm (15.4 mm) 3 rd Floor 60.7 mm (21.5 mm) 59.9 mm (20.8 mm) 33.2 mm (0.3 mm) 2 nd Floor 39.2 mm (21.1 mm) 39.1 mm (21.0 mm) 32.9 mm (15.5 mm) 1 st Floor 18.1 mm (18.1 mm) 18.1 mm (18.1 mm) 17.4 mm (17.4 mm) Ground0 mm 27

28. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Optimization Tips… 1.Bracing designs that reduce shear forces in the columns will be most successful 2.Efficiency over Stiffness  Form comparable stiffness ratios, chose option with higher Efficiency 3.Story with largest interstory drift needs stiffest bracing allowed by constraints  Initially, this is the mid-height of the tower, but will change with addition of bracing 4.Bracing of mid-level floors should be most effective  Reduces drift of neighboring floors as well Betting on a Design 28

29. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Summary of Tower Design Problem Limit states –allowed deflection at 4.5 N applied load (stiffness) Constraints –where can bracing be added –also tower dimensions, required K’nex connections, etc. Efficiency –cost of bracing for stiffness achieved Betting on a Design29 bracing Determine a bracing scheme for the tower Each team has different limit states & constraints