EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Design optimization: optimization problem and factor of safety (F.O.S.) Introduction to Engineering Systems Lecture 8 (9/21/2009) Prof. Andrés Tovar

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Announcements 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. 2Optimizing your design

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame From last class Cost and constraints Efficiency E=S/B –Stiffness ratio S=k br /k ubr =  ubr /  br, –Bracing ratio B=L br /L ubr Shearing and overturning (bending): shear and axial actions Interstory drift:  interstory =  ceiling –  floor Cumulative drift (total deflection at each floor): sum of all previous  interstory Four design optimization tips: –Reduce shear in columns –Maximize efficiency –Brace floor with the largest  interstory –Brace intermediate floors 3Optimizing your design

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Where we are in the tower design process 4 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 Optimizing your design

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame What is optimization? 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. 5 Structures of maximum stiffness and minimum weight Optimizing your design

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Ingredients of an optimization problem 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. 6Optimizing your design

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Optimizing a parabolic shot Problem formulation –Given v 0 (constant parameter) –Find the angle  (design variable) –that maximizes the distance D (objective function) –subject to 0 ≤  ≤ 90 (constraint) 7  D v0v0 Optimizing your design

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Graphic method 8 System  D Optimizing your design

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame 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 9Optimizing your design

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Optimizing the tower building What are your design variables? What is your objective function? What are your constraints? What optimization method are you going to use? 10Optimizing your design

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame 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)  max  Tower ≤  max (ideal, deterministic world)   ≤ (F.O.S.) ×  max (real, probabilistic world) 11Optimizing your design

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame 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 12Optimizing your design

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Use of Factor of Safety in the design process 13 Displacement (mm) Force (N) 4.5  max  SAP k min k SAP Optimizing your design

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame From experimental and theoretical data to F.O.S. 14 Displacement (mm) Force (N) 4.5  exp  SAP  k = 0.07 N/mm  k = 0.02 N/mm k SAP = N/mm Optimizing your design

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Factor of Safety and Tower Stiffness: Remarks stiffness of proposed design in SAP2000 mean of stiffness from trials standard deviation 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 1) Determine F.O.S. from theoretical (SAP200) and experimental models Optimizing your design15

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Example Given  k =0.07N/mm,  k =0.02N/mm, and N  =2, determine the F.O.S. and the target displacement in the SAP model if the limit state is  max =10.0mm. 1) Determine a F.O.S. 2) Determine the target displacement in you SAP model 16Optimizing your design

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Example Given  k =0.07N/mm,  k =0.02N/mm, and N  =2, determine the F.O.S. and the target displacement in the SAP model if the limit state is  max =10.0mm. What if your final  SAP = 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 17  Optimizing your design

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Using Matlab to determine probability of failure 18 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

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Prediction worksheet UNBRACED TOWER RECAP SAP Stiffness (units?): 0.08 N/mm Mean Stiffness from Experimental Database (units?): 0.07 N/mm 19Optimizing your design

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Prediction worksheet BRACED TOWER Target Deflection Limit State (units?): 10 mm (this is an example) Factor of Safety: 2.7 (for 2  design, usual values are between 1.5 and 2 for this project) SAP Predicted displacement under 4.5 N load (units?): mm Calculated Bracing Ratio for Tower (B): (848.5mm/3000mm) SAP Predicted Stiffness Ratio (S): (101.1mm/67.1mm) SAP Predicted Efficiency Ratio (E = S/B): (1.5067/0.2828) 20Optimizing your design

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Optimization evaluation worksheet 21Optimizing your design Stiffness ratio S=k br /k ubr =  ubr /  br S= (101.1mm/67.1mm) Bracing ratio B=L br /L ubr B= (848.5mm/3000mm) Efficiency Ratio E=S/B E= (1.5067/0.2828)

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Results from structural optimization software NDopti 22Optimizing your design

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Results from structural optimization software NDopti 23Optimizing your design