MITESD_77S10_soln05 (1)

MITESD_77S10_soln05 (1) - 16.888/ESD 77 Multidisciplinary...

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16.888/ESD 77 Multidisciplinary System Design± Optimization:± Assignment 5 Part a) Solution± Part A1 Recap from A4: This problem relates to A4 and the optimal solutions obtained by using SQP exhaustively for all possible discrete variable settings is shown in table below: Problem formulation with SQP b (m) t (m) h (m) w (m) d (m) Beam materi al Support Material # beams Optimal Cost w/o shear stress constraint 0.3471 0.01 2.0 0.374 0.374 A514 Concrete 1 $5771.3 With constraint linking beam width (b), number of beams (N) and support depth (d) 0.3471 0.01 2.0 0.3664 0.3971 A514 Concrete 1 $5774.1 We will use the optimal vector thus obtained to answer all subsequent questions, wherever applicable. (a) Here the applied load is given as: F = q*L and q (load per unit length) becomes an ± additional continuous design variable. For calculation of the Jacobian, we have to leave± out the discrete variables like material id’s and number of beams. ± The Jacobian with respect to continuous design variables can be written as:± T 2 At 0 A (2 b + h 4 ) t 0 C F T At 0 Jacobian = = where X = [ b t h w d q ]'; A = c L ρ n ; B = c H b b b s s X X Bd 0 Bw 0 0 L Using the given data, we have, A=213.3e+3 ($/m 2 ) for beam and B=0.48e+03 ($/m 2 ) for concrete column. Note this analysis will also depend on your problem formulation in A4 and would be carried over while tackling this question.
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Considering the formulation laid out in A4, the Jacobian for the case where no shear stress constraint and no constraint on internal dependency of beam flange width with support column depth (d) is given by: T 4266 0 566141 0 2133 0 Jacobian = 163.2 0 180 0 0 30 For the other case where we consider shear stress constraint and also a constraint on internal dependency of beam flange width (b) and column depth(d), the Jacobian is: T 4266 0 566141 0 2133 0 Jacobian = 191 0 176.4 0 0 30 Notice the sensitivity of cost to the beam thickness (t). Also notice the changes in
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This note was uploaded on 11/08/2011 for the course AERO 16.851 taught by Professor Ldavidmiller during the Fall '03 term at MIT.

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MITESD_77S10_soln05 (1) - 16.888/ESD 77 Multidisciplinary...

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