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Optimal 3-member Truss Design

Optimal 3-member Truss Design. Submitted By :- Course Instructor :- Avinash Kumar (10105017) Prof. K. Deb Piyush Rai (10105070) (ME 752). Coming Next …. Problem Formulation Objective function definition Constraints definition

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Optimal 3-member Truss Design

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  1. Optimal 3-member Truss Design Submitted By :- Course Instructor:- Avinash Kumar (10105017) Prof. K. Deb PiyushRai (10105070) (ME 752)

  2. Coming Next …. Problem Formulation Objective function definition Constraints definition KKT conditions formulation Optimal Solution to the problem Penalty function method Augmented Lagrangian Method Results

  3. L L 3 Problem Formulation 1 A2 2 A3 A1 L δ1 S2 = Pcosθ Fig. 1. A 3- member truss. θ P δ2 S1 =Psinθ

  4. Objective Stress constraints Displacement constraints minimize the weight of the structure (W) optimal values of cross-sectional areas A1 , A2 , A3 Minimize weight,w = f(A1 , A2 , A3) subjected to g1 : σ1 ≤ σ1 (allowable) g2 : σ2 ≤ σ2 (allowable) g3 : σ3 ≤ σ3 (allowable) g4 : δ1≤ δ1 (allowable) g5 : δ2≤ δ2 (allowable) and Ai ≥ 0 ; i= 1,2,3.

  5. Mathematical formulation Material properties :- Mild steel (ρ = 7860 kg/m3 ,E=2.1*1011 Pa) Taking L=10 m , P = 50 KN, Max. allowable stress, σall. = 220 MPa, δ1 (all.)=δ2 (all.) = 50mm. Objective function : Minimize W = ρg(10√2 A1+10A2+10√2 A3) subjected to,

  6. KKT Conditions 1)(Equilibrium Equation) Finding out the gradients of objective function & all constraints , we get :- where, ,

  7. 2) ujgj(x)=0 (Complementary Slackness Condition) u1g1(x)=0 u2g2(x)=0 u3g3(x)=0 u4g4(x)=0 u5g5(x)=0 3) uj ≥ 0 ; j = 1,2,…..,5. (Non-negativity )

  8. Optimal Solution to the problem Comparison of Penalty function method and Augmented Lagrangian Method :- R=1 For an initial value of R=1, Augmented Lagrangian method is better than penalty function method. The optimal solution we get :- A1 = 0.1323 m2 , A2 = 0.0492 m2 , A3 = 0.000494 m2 Minimum weight of the structure, Wmin. = 19.155 tonnes.

  9. R=10 Here also, in terms of function evaluations, Augmented lagrangian method is better than the Penalty function method. The optimal solution for R=10 is :- A1 = 0.2123 m2 , A2 = 0.03874 m2 , A3 = 0.03179 m2 Minimum weight of the structure, Wmin. = 28.74 tonnes.

  10. Results For the 3-member structure, and for the selected values of maximum allowable stress, applied loads,etc., the Augmented Lagrangian method is better than Penalty function method. For a large no. of variables, Penalty function method is better than Augmented Lagrangian method in terms of no. of function evaluations.

  11. References Haug & Arora, Applied Optimal Design . K.Deb, Optimization for Engg. Design.

  12. THANK YOU

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