Design optimization: metrics and tower bracing

1. Design optimization: metrics and tower bracing Introduction to Engineering Systems Lecture 7 (9/18/2009) Prof. Andrés Tovar

2. 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 Design

3. Where’s the Bug? 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 Betting on a Design

4. Where’s the Bug? 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 Betting on a Design

5. Where’s the Bug? x = 1:10; a = 2; b = 3; c = 5; y = a*x^2 + b*x + c Betting on a Design

6. Where’s the Bug? 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) Betting on a Design

7. Where’s the Bug? “mean” is a predefined MATLAB function name. Don’t use it as a variable name!!! mean = sum(x)/length(x) c_o_v = std(x)/mean(x) Note that “cov” is also a MATLAB function name. (Calculates something called a “covariance”) Betting on a Design

8. Testing for and Counting Values in Range see Ch 10 and Video 11 >> x = [1 3 5 7 9] x = 1 3 5 7 9 >> x < 6 ans = 1 1 1 0 0 >> (x < 6) & (x >= 3) % "&" means "and" ans = 0 1 1 0 0 >> sum((x < 6) & (x >= 3)) % use sum to count ans = 2 Betting on a Design

9. Histograms see Ch 10 and Video 12 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) Use arguments to hist to control number of “bins” See also histc to get bin counts Betting on a Design

10. From last class P(16≤d≤18) = 0.55 (experimental) P(16≤d≤18) = 0.53 (theoretical) Normal distribution (PDF) 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 Design

11. From last class p= 0.5280, a = 1 q = 0.2016, b = -5 Expected gain (G) based on probability (Huygens’ formula) Betting on a Design

12. Probability of failure m m+s m+2s m+3s Betting on a Design

13. Design considerations Limit state: (Civil Structural Engineering) objective on the performance or behavior of a design, e.g., Survivability, Serviceability, Habitability. • Efficiency • E = S/B • S = kbr/kubr: stiffness ratio • B = Lbr/Lubr: bracing ratio Betting on a Design

14. 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 Design

15. Constraints Pet architecture, Nayoga, Japan • 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 Design

16. Tower Metrics: Bracing Ratio Factor by which bracing increases tower cost. Ratio of length of braces addedLbr to the design divided by the total length of vertical members Lubrin the design where Lubr = 4 columns/floor x N floors x 150 mm/column. Betting on a Design

17. Tower Metrics: Stiffness Ratio Factor by which bracing improves tower stiffness Betting on a Design

18. Efficiency Improvement in stiffness relative to cost. Higher efficiency leads to a better design. Betting on a Design

19. Bracing Scheme Options Design 2 Design 3 Design 1 Design 5 Design 4 Design 6 (3-X braces)

20. Optimization of Tower Design • Need to First Understand: • How bracing affects our displacement and why? • How do different bracing schemes affect efficiency? • How does bracing placement affect our displacement & efficiency?

21. Load Paths of Unbraced Structure Shearing

22. Load Paths of our Braced Structure Minimizes Shear Minimizes Deflections What types of bracing do you think will decrease the stiffness the most?

23. Tower Behavior Under Specified Loading Does the tower represent the Axial or Shear Action Behavior? How can we use this information?

24. Optimization Tips… • Bracing designs that reduceshear forces in the columns will be most successful • Efficiency over Stiffness • Form comparable stiffness ratios, chose option with higher Efficiency

25. 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

26. Bracing Location Braced @ 3rd Floor Unbraced Braced @ Top Floor

27. What about Location of Bracing and Drift?

28. Optimization Tips… • Bracing designs that reduceshear forces in the columns will be most successful • Efficiency over Stiffness • Form comparable stiffness ratios, chose option with higher Efficiency • 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 • Bracing of mid-level floors should be most effective • Reduces drift of neighboring floors as well Betting on a Design

29. Summary of Tower Design Problem • Determine a bracing scheme for the tower bracing • Each team has different limit states & constraints • 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 Design