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Design optimization: optimization problem and factor of safety (F.O.S.)

Design optimization: optimization problem and factor of safety (F.O.S.). Introduction to Engineering Systems Lecture 8 (9/21/2009). Prof. Andrés Tovar. Announcements. Optimizing your design. HW3 and LC deliverable is due this week. Print and read your LC material before class.

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Design optimization: optimization problem and factor of safety (F.O.S.)

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  1. Design optimization:optimization problem and factor of safety (F.O.S.) Introduction to Engineering Systems Lecture 8 (9/21/2009) Prof. Andrés Tovar

  2. Announcements Optimizing your design HW3 and LC deliverable is due this week. Print and read your LC material before class. HW4 will be posted today. Print out HW4 (the technical memo), read it, highlight any problems, and bring it to class on Friday for an in-class exercise.

  3. From last class Optimizing your design • Cost and constraints • Efficiency E=S/B • Stiffness ratio S=kbr/kubr=Dubr/Dbr, • Bracing ratio B=Lbr/Lubr • Shearing and overturning (bending): shear and axial actions • Interstory drift: Dinterstory = Dceiling–Dfloor • Cumulative drift (total deflection at each floor): sum of all previous Dinterstory • Four design optimization tips: • Reduce shear in columns • Maximize efficiency • Brace floor with the largest Dinterstory • Brace intermediate floors

  4. 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 Optimizing your design

  5. Optimizing your design What is optimization? Structures of maximum stiffness and minimum weight The word optimization comes from the Latin optimum which means the best. Optimize is finding the design variables that maximize or minimize an objective function subject to a set constraints.

  6. Ingredients of an optimization problem Optimizing your design • Design variables: an engineering quantity to be determined during the optimization procedure. • The set of all design variables defines the design space. • A constant parameter is a value that cannot be modified. • Objective function: a function to be maximized or minimized. • Design constraints: express the limits in performance or limits in the design variables. • Limits in performance are expressed as functions and referred to as functional constraints. • Limits in the design variables are referred to as geometric constraints. • The set of design variable satisfying all constraints is referred to as feasible space.

  7. Optimizing a parabolic shot v0 q D Optimizing your design • Problem formulation • Given v0 (constant parameter) • Find the angle q (design variable) • that maximizes the distance D (objective function) • subject to 0 ≤ q≤ 90 (constraint)

  8. Graphic method q D System Optimizing your design

  9. Optimizing your design Some optimization methods • Graphic optimization • Analytic methods: mathematical analysis of optimality conditions • Numerical methods: use of computer • Gradient-based methods: compute derivatives • Direct methods: do not compute derivatives • Deterministic: such as exhaustive search • Probabilistic: such as genetic algorithms • Heuristic: based on experience

  10. Optimizing the tower building Optimizing your design What are your design variables? What is your objective function? What are your constraints? What optimization method are you going to use?

  11. Optimizing your design Proposed optimization problem • Find the bracing scheme • That maximized the efficiency E=S/B of the tower • Subject to • Bracing constraints due to the use of the floors • Symmetry condition on bracing • Target deflection limit state (mm) Dmax DTower≤ Dmax(ideal, deterministic world) mD≤ (F.O.S.)×Dmax(real, probabilistic world)

  12. Optimizing your design Factor of Safety • Variation and uncertainty  too risky to design to satisfy limit states exactly • Overdesign product to be confident that limit states will be satisfied • but overdesign increases cost • Factor of safety: how much should we overdesign • to be sufficiently confident that limit states will be satisfied • with minimal additional cost

  13. Optimizing your design Use of Factor of Safety in the design process Force (N) kSAP kmin 4.5 Dmax DSAP Displacement (mm)

  14. Optimizing your design From experimental and theoretical data to F.O.S. kSAP = 0.081 N/mm Force (N) mk = 0.07 N/mm sk = 0.02 N/mm 4.5 Dexp DSAP Displacement (mm)

  15. Optimizing your design Factor of Safety and Tower Stiffness: Remarks 1) Determine F.O.S. from theoretical (SAP200) and experimental models stiffness of proposed design in SAP2000 standard deviation of stiffness from trials mean of stiffness from trials how many  to achieve desired confidence? (3 for 99.9%) 2) Use F.O.S. to set your state limits on SAP200

  16. Optimizing your design Example Given mk=0.07N/mm, sk=0.02N/mm, and Ns=2, determine the F.O.S. and the target displacement in the SAP model if the limit state is Dmax=10.0mm. 1) Determine a F.O.S. 2) Determine the target displacement in you SAP model

  17. Optimizing your design Example m+1.88s Given mk=0.07N/mm, sk=0.02N/mm, and Ns=2, determine the F.O.S. and the target displacement in the SAP model if the limit state is Dmax=10.0mm. What if your final DSAP = 4mm, would you accept the design? What would be the effective F.O.S.? Let us determine the effective F.O.S. Therefore, the probability of failure will be given by As F.O.S. increases so does material may reduce EFFICIENCY

  18. Using Matlab to determine probability of failure m = 0.07; % mean (mu) s = 0.02; % standard deviation (sigma) N = 1.88; % number of sigmas from mu % theoretical probability of failure pf = cdf('norm', m-N*s, m, s) m m-N*s Optimizing your design

  19. Optimizing your design Prediction worksheet UNBRACED TOWER RECAP SAP Stiffness (units?): 0.08 N/mm Mean Stiffness from Experimental Database (units?): 0.07 N/mm

  20. Optimizing your design Prediction worksheet BRACED TOWER Target Deflection Limit State (units?): 10 mm (this is an example) Factor of Safety: 2.7 (for 2s design, usual values are between 1.5 and 2 for this project) SAP Predicted displacement under 4.5 N load (units?): 101.10 mm Calculated Bracing Ratio for Tower (B): 0.2828 (848.5mm/3000mm) SAP Predicted Stiffness Ratio (S): 1.5067 (101.1mm/67.1mm) SAP Predicted Efficiency Ratio (E = S/B): 5.3270 (1.5067/0.2828)

  21. Optimizing your design Optimization evaluation worksheet • Stiffness ratio • S=kbr/kubr=Dubr/Dbr • S=1.5067 (101.1mm/67.1mm) • Bracing ratio • B=Lbr/Lubr • B=0.2828 (848.5mm/3000mm) • Efficiency Ratio • E=S/B • E=5.3270 (1.5067/0.2828) 67.1 0.0 18.5 48.6 48.6 0.0 15.4 33.2 33.2 0.3 848.5 32.9 32.9 15.5 0.0 17.4 17.4 0.0 17.4 0.0

  22. Results from structural optimization software NDopti Optimizing your design

  23. Results from structural optimization software NDopti Optimizing your design

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