EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Statistics and Probability: Betting on a Design Introduction to Engineering Systems Lecture 5 (9/11/2009) Prof. Andrés Tovar

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Announcements SAP2000 (Windows only) is now available on CD in the Engineering Library Avoid printing in the Learning Center, especially class handouts! 2Betting on a Design

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Betting on a Tower Design SAP2000 simulates ideal tower behavior –see the effects of changing the bracing without having to build it. Betting on a Design3 But the real world isn’t ideal Use SAP2000 (theoretical model) together with experimental data (empirical model) to design a bracing scheme that is safe (limited displacement) but that is still cost efficient Where should we aim? key idea SAP2000

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Empirical model Betting on a Design % The launcher X = (0.25:0.25:1.5)'; D = [1;4;10;18;27;36]; % linear prediction c = polyfit(X,D,1); Dc = polyval(c,X); % cuadratic prediction d = polyfit(X,D,2); Dd = polyval(d,X); plot(X,D,'ro-',X,Dc,'b-',X,Dd,'k--') xlabel('horizontal pullback X (m)') ylabel('distance D (m)') title('the launcher') piecewise-linear model (ro-) linear model (b-) quadratic model (k--) Predictions depend on the basis function of the model.

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. The bet: can you come within 1 m of a target at a distance D = 18 m? Betting on a Design6 X D

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame How consistent is the launcher? Suppose that on Monday you determined an optimal value for X (from experimentation) to launch a ball 18 m On Tuesday, you ran 20 trials at that same setting for X and got the following results: Betting on a Design7

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Review: Model Inputs and Outputs Design variables (input, independent) –pullback X Environmental variables –wind velocity (speed and direction) –temperature –name some more… Behavioral variables (output, dependent) –distance D Betting on a Design8

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Sources of Uncertainty in a Model Betting on a Design9 Uncertainty can be 1)Epistemic lack of knowledge 2)Aleatory random events

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Scatterplot of Trials Betting on a Design10

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Arithmetic Mean Usually just called the “mean” (but there are other kinds of means) What is usually meant when we say “average” (but there are other kinds of averages) Betting on a Design11 In Matlab see function mean. Out of curiosity you can check median and mode.

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Mean and Error About the Mean Betting on a Design12

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Standard Deviation Square root of... the average of the squares of the errors Betting on a Design typo in equation (5.4) In this Class: = 1.4 m In Matlab see function std

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Coefficient of Variation Standard deviation divided by the mean Normalized version of the standard deviation Betting on a Design14

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Systematic and Random Error Systematic error –error between the mean of the trials and the target Random error –random error of trials about the mean Betting on a Design target Why did I miss the target? 15

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Causes of Systematic and Error How many can you think of? Systematic error –Wind between two days –Temperature –Settings, e.g., angles, references Random error –Wind while the softball is flying –Grip of the softball –Distractions Betting on a Design16

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Back to the Tower Betting on a Design17 Displacement Scatter plot Hooke’s Law k = stiffness coefficient

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Displacement statistics Betting on a Design  ––  18

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Best-Fit Tower Force-Displacement Lines Betting on a Design19

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Stiffness statistics Betting on a Design20  ––   = 0.07 N/mm  = 0.02 N/mm

Understanding Excel Spreadsheet of Test Results (A typical section tab) Dial gauge measurements from Section 1 Red test fixture. Applied Forces Measurement after initial displacement is subtracted. Stiffness as computed by Excel Function (includes stiffness of dial gauge). Corrected Tower stiffness (after removal of dial gauge) Results from yellow test fixture.

The Summary Tab of the Excel Spreadsheet Table containing tower stiffness values for each tower tested in Learning Center 1 (ktower from Section tabs) The mean and standard deviation of the entries in the Summary Table

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame How to determine an equivalent stiffness? Betting on a Design Springs in series

EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame How to determine an equivalent stiffness? Betting on a Design Springs in parallel

Team 1 Floor 1 Floor 4 Floor 2 Floor 3 Floor 5 Background / Usage: Mixed usage. Urban setting with the first 2 floors designated as parking garage so no interior bracing on those floors. The top 3 floors are for a hotel, with the top most floor (floor 5) being the penthouse so no bracing of any kind on that floor. Floors 3 & 4 cannot have any exterior diagonals. Parking Hotel Penthouse of Hotel Limit State (under 4.5N load): 5 mm Note: No bracing on a floor still Allows the addition of the center Plane vertical members

Team 2 Background / Usage: Mixed usage. Located near a college campus, the first 3 floors are student apartments (any bracing scheme is permitted). The top 2 floors are high end loft style condominiums that require wide open spacing and unobstructed views (no bracing is permitted). Floor 1 Floor 4 Floor 2 Floor 3 Floor 5 Apt. Condo Apt. Note: No bracing on a floor still Allows the addition of the center Plane vertical members Limit State (under 4.5N load): 19 mm Example: Eddy Street Commons – South Bend, IN

Team 3 Background / Usage: Mixed usage. Urban setting with the bottom floor being reserved for store front, the user of which has not been determined. In order to maximize flexibility no diagonal bracing is permitted on exterior of the first floor. The middle 3 floors are financial office suites, any bracing is permitted. Finally the top floor is for a Steakhouse, so no exterior bracing of any kind on that floor. Floor 1 Floor 4 Floor 2 Floor 3 Floor 5 Office Restaurant Store Front Office Limit State (under 4.5N load): 7 mm

Team 4 Background / Usage: Mixed usage. The bottom 2 floors are being used by Dave & Busters, and will not allow any interior bracing. The top 3 floors are office space, with the CEO’s offices taking up the middle floor. In order to have flexibility for unobstructed views of the city, there is no exterior bracing permitted on the middle floor. Floor 1 Floor 4 Floor 2 Floor 3 Floor 5 CEO Office Dave & Busters Dave & Busters Note: No bracing on a floor still Allows the addition of the center Plane vertical members Limit State (under 4.5N load): 5 mm

Team 5 Background / Usage: Mixed usage. The bottom 2 floors are for a theater – they have very few exterior windows so exterior bracing is permitted but no interior bracing. The hotel requires no exterior diagonal bracing (any other form is permitted) Floor 1 Floor 4 Floor 2 Floor 3 Floor 5 Hotel Theater Hotel Theater Limit State (under 4.5N load): 10 mm

Team 6 Background / Usage: This building is for IKEA. The first floor does not allow any bracing so store front windows would be unobstructed and there are open areas for displays. Likewise, the top 4 floors are not permitted to have any interior diagonal bracing (any form of exterior bracing is permitted). Floor 1 Floor 4 Floor 2 Floor 3 Floor 5 IKEA Store Front IKEA Limit State (under 4.5N load): 8 mm Note: No bracing on a floor still Allows the addition of the center Plane vertical members

Team 7 Floor 1 Floor 4 Floor 2 Floor 3 Floor 5 Background / Usage: This building is a hospital. All floors require minimum obstructions for easy maneuvering & observation, so no interior bracing is allowed. The first floor is for outpatients and is the location of the emergency room- this requires easy access, so no diagonal bracing is allowed on this floor. All other floors allow any exterior bracing scheme. Limit State (under 4.5 N load): 4 mm Outpatient & ER Hospital