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Pratt truss optimization using

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Harish Kant Soni

The document describes optimizing the design of a Pratt truss bridge using a genetic algorithm. It presents the objective of minimizing the total cross-sectional area of the truss members subject to constraints on member size and displacement. The genetic algorithm is applied over multiple generations using selection, crossover and mutation to iteratively evolve solutions toward an optimal design. The results show the optimized cross-sectional area in cm^2 for each of the 22 truss members.

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Pratt Truss Optimization Using Genetic Algorithm By :- Harish Kant Soni Roll No:- 12CE31004
Use of Pratt truss Fig :- Gatton Railway Bridge, Queensland, Australia Image Source :- https://commons.wikimedia.org/wiki/File:Gatton_Railway_Bridge.JPG
Forces in Pratt truss Image source :-http://www.slideshare.net/sheikhjunaidyawar/trusses-28955458 F
18 m 7 m
Loading Train’s engine has highest weight as compared to other coaches Indian locomotive class WAP-5 • Length = 18.16 m • Weight = 79251.7 kg Source :- https://en.wikipedia.org/wiki/Indian_locomotive_class_WAP-5
18 m 7 m
18 m 7 m Dead Load = 150 kN/ meter/ side Total dead load = 150 x 18 = 2700 KN Live load = 800 KN 550 KN 550 KN 550 KN 550 KN 550 KN 1 2 3 4 5 6 7 12 11 10 9 8
Objective Function:- min. total Area = min A(1)+A(2)+A(3)+A(4)+A(5)+A(6)+A(7)+A(8) +A(9)+A(10)+A(11)+A(12)+A(13)+A(14)+A(15)+ +A(16)+A(17)+A(18)+A(19)+A(20)+A(21)+A(22) Constraints :- min. Area = 0.0010 m^2 = 10 cm^2 disp at any node < 50 mm
clc; clear all; % initial cromosome having area in the range 0.002 to 0.02 m^2- C = 2e-3*[1 2 4 10 3 9 10 7 1 5 1 1 2 4 10 3 9 10 7 1 5 2; 9 3 7 9 4 7 5 2 4 9 1 9 3 7 9 4 7 5 2 4 9 3; 7 3 4 1 1 7 3 2 4 9 8 7 3 4 1 1 7 3 2 4 9 4; 1 5 7 9 4 9 5 2 4 9 6 1 5 7 9 4 9 5 2 4 9 5; 4 10 6 2 2 2 3 2 4 1 9 4 10 6 2 2 2 3 2 4 1 6; 9 3 5 9 5 7 5 1 4 9 1 9 3 5 9 5 7 5 1 4 9 7; 9 9 4 1 4 3 1 2 4 9 8 9 9 4 1 4 3 1 2 4 9 8; 1 3 7 9 1 7 5 5 4 4 1 1 3 7 9 1 7 5 5 4 4 9; 9 8 3 1 2 4 2 2 4 9 6 9 8 3 1 2 4 2 2 4 9 10; 2 3 7 9 3 7 5 9 4 3 1 2 3 7 9 3 7 5 9 4 3 1; 8 7 2 10 4 5 3 10 4 7 4 8 7 2 10 4 5 3 10 4 7 2; 1 2 4 10 3 9 10 7 1 5 1 1 2 4 10 3 9 10 7 1 5 3; 9 3 7 9 4 7 5 2 4 9 1 9 3 7 9 4 7 5 2 4 9 4; 7 3 4 1 1 7 3 2 4 9 8 7 3 4 1 1 7 3 2 4 9 5; 1 5 7 9 4 9 5 2 4 9 6 1 5 7 9 4 9 5 2 4 9 6; 4 10 6 2 2 2 3 2 4 1 9 4 10 6 2 2 2 3 2 4 1 7; 9 3 5 9 5 7 5 1 4 9 1 9 3 5 9 5 7 5 1 4 9 8; 9 9 4 1 4 3 1 2 4 9 8 9 9 4 1 4 3 1 2 4 9 9; 1 3 7 9 1 7 5 5 4 4 1 1 3 7 9 1 7 5 5 4 4 10; 9 8 3 1 2 4 2 2 4 9 6 9 8 3 1 2 4 2 2 4 9 1; 2 3 7 9 3 7 5 9 4 3 1 2 3 7 9 3 7 5 9 4 3 2; 2 3 7 9 3 7 5 9 4 3 1 2 3 7 9 3 7 5 9 4 3 3];
F_obj= sum(C,2); % return sum of rows in a column matrix of 22 X 1 %% selection Fitness = zeros(22,1); for a = 1:22 Fitness(a) = (1/(1+F_obj(a))); end S = sum(Fitness); Prob_of_cromosome = Fitness/S; roullet = zeros(22,1); roullet(1) = Prob_of_cromosome(1); for a = 2:22; roullet(a) = roullet(a-1) + Prob_of_cromosome(a); %CDF end
%% new set of cromosome for a = 1 : 22 r = rand; if (r <= roullet(1)) new_cromosome_id = 1; else for b = 1 : 21 if (r > roullet(b) & r <= roullet(b+1)) new_cromosome_id = b+1; end end end C(a,:) = C (new_cromosome_id,:); end
%% Cross Over C_new = C; for a= 1:22 r_crossover = rand(); if(r_crossover < 0.7) r_location = 10*round(rand(),1); % generates random number between 0 to 10. It selects location for crossover for b = r_location+1 : 22 if (a==22) C_new(a,b)= C(1,b); C_new(1,b) = C(22,b); else C_new(a,b)= C(a+1,b); C_new(a+1,b) = C(a,b); end end end end
%% Mutation Probability 0.1 for a=1:22 for b=1:22 if(rand < 0.1) C_new(a,b)= round(((0.02-0.001)*rand+0.001),3); end end end
%% Truss code coordinate=[0 0 ;3 0 ;6 0;9 0; 12 0; 15 0; 18 0;15 7; 12 7; 9 7; 6 7; 3 7;]; connectivity=[1 2;2 3;3 4;4 5;5 6; 6 7; 7 8;8 9; 9 10;10 11; 11 12;1 12; 2 12;3 12;3 11; 4 11;4 9; 4 10; 5 9; 5 8;6 8; 7 8]; boundary=[1 1 ; 0 0 ; 0 0 ;0 0 ;0 0;0 0; 1 1;0 0; 0 0;0 0;0 0;0 0]; load=[0 0 0 -550e3 0 -550e3 0 -550e3 0 -550e3 0 -550e3 0 0 0 0 0 0 0 0 0 0 0 0]; Elasticity=2.E+11*[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1];
%% Penalty abs_unknown_displacement = abs(unknown_displacement); count = 0; for k = 1 : 22 for l = 1:20 if(abs_unknown_displacement(k,l)-0.050 > 0) count = count + 1; end C_new(k,:)= (1+0.05*count)*C_new(k,:); end end
Results
Results
member 1 2 3 4 5 6 7 8 9 10 11 0.019 0.004 0.016 0.015 0.019 0.008 0.005 0.008 0.012 0.014 0.019 0.019 0.015 0.016 0.003 0.009 0.015 0.005 0.008 0.003 0.014 0.003 0.019 0.015 0.016 0.017 0.009 0.008 0.005 0.006 0.01 0.008 0.019 0.019 0.015 0.016 0.003 0.005 0.014 0.005 0.011 0.003 0.014 0.003 0.019 0.004 0.016 0.015 0.019 0.019 0.005 0.006 0.01 0.008 0.015 0.019 0.015 0.016 0.003 0.005 0.016 0.013 0.008 0.003 0.014 0.003 0.019 0.011 0.016 0.003 0.005 0.016 0.009 0.008 0.003 0.014 0.003 0.019 0.004 0.016 0.003 0.005 0.016 0.013 0.017 0.003 0.003 0.003 0.019 0.015 0.016 0.015 0.019 0.008 0.005 0.008 0.012 0.014 0.003 0.019 0.004 0.016 0.019 0.011 0.008 0.005 0.006 0.01 0.008 0.005 0.008 0.015 0.016 0.015 0.019 0.008 0.005 0.008 0.016 0.014 0.003 0.019 0.015 0.016 0.009 0.005 0.016 0.013 0.008 0.003 0.014 0.003 0.006 0.004 0.016 0.015 0.019 0.008 0.005 0.008 0.012 0.014 0.016 0.019 0.015 0.016 0.003 0.005 0.016 0.013 0.008 0.003 0.014 0.016 0.019 0.015 0.016 0.007 0.019 0.014 0.019 0.008 0.019 0.014 0.003 0.019 0.002 0.016 0.015 0.005 0.016 0.013 0.008 0.003 0.014 0.003 0.019 0.004 0.016 0.015 0.019 0.014 0.005 0.008 0.019 0.014 0.003 0.019 0.015 0.016 0.003 0.005 0.015 0.005 0.008 0.003 0.016 0.003 0.019 0.015 0.006 0.003 0.009 0.016 0.013 0.008 0.003 0.014 0.003 0.019 0.015 0.009 0.017 0.009 0.002 0.009 0.009 0.01 0.015 0.016 0.019 0.015 0.016 0.014 0.005 0.016 0.013 0.008 0.003 0.014 0.003 0.019 0.015 0.016 0.003 0.005 0.016 0.013 0.008 0.003 0.014 0.003 value 0.019 0.015 0.016 0.003 0.005 0.015 0.005 0.008 0.003 0.014 0.003 times 20 14 20 9 6 14 10 16 12 17 15
12 13 14 15 16 17 18 19 20 21 22 0.008 0.014 0.016 0.007 0.002 0.018 0.016 0.014 0.01 0.006 0.002 0.008 0.014 0.012 0.007 0.002 0.011 0.016 0.008 0.017 0.006 0.002 0.004 0.014 0.012 0.016 0.002 0.013 0.017 0.003 0.016 0.012 0.006 0.008 0.014 0.012 0.008 0.002 0.018 0.016 0.014 0.01 0.006 0.008 0.008 0.014 0.012 0.007 0.002 0.018 0.016 0.014 0.01 0.006 0.002 0.008 0.014 0.012 0.007 0.002 0.012 0.016 0.014 0.02 0.006 0.002 0.008 0.014 0.019 0.007 0.002 0.018 0.016 0.014 0.01 0.019 0.002 0.008 0.014 0.012 0.007 0.002 0.018 0.016 0.014 0.01 0.006 0.002 0.008 0.014 0.016 0.007 0.002 0.018 0.016 0.014 0.017 0.006 0.002 0.004 0.014 0.012 0.016 0.002 0.012 0.017 0.003 0.016 0.012 0.006 0.008 0.014 0.002 0.007 0.002 0.018 0.016 0.014 0.01 0.006 0.002 0.008 0.02 0.016 0.001 0.002 0.018 0.016 0.014 0.01 0.006 0.002 0.008 0.019 0.017 0.007 0.002 0.018 0.019 0.014 0.01 0.006 0.002 0.008 0.014 0.012 0.007 0.002 0.018 0.016 0.014 0.01 0.006 0.002 0.008 0.001 0.012 0.007 0.002 0.018 0.011 0.014 0.01 0.006 0.002 0.008 0.014 0.012 0.007 0.002 0.018 0.016 0.014 0.01 0.006 0.002 0.008 0.014 0.012 0.007 0.004 0.018 0.016 0.014 0.01 0.006 0.002 0.008 0.014 0.012 0.007 0.002 0.018 0.016 0.008 0.013 0.006 0.002 0.008 0.014 0.012 0.007 0.002 0.018 0.016 0.014 0.01 0.006 0.002 0.008 0.015 0.012 0.016 0.002 0.002 0.017 0.017 0.007 0.012 0.002 0.008 0.014 0.012 0.007 0.014 0.018 0.016 0.014 0.01 0.006 0.002 0.008 0.014 0.012 0.006 0.002 0.004 0.016 0.014 0.01 0.006 0.002 0.008 0.014 0.012 0.007 0.002 0.018 0.016 0.014 0.01 0.006 0.002 20 19 17 18 21 16 20 16 15 18 19

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Pratt truss optimization using

  • 1. Pratt Truss Optimization Using Genetic Algorithm By :- Harish Kant Soni Roll No:- 12CE31004
  • 2. Use of Pratt truss Fig :- Gatton Railway Bridge, Queensland, Australia Image Source :- https://commons.wikimedia.org/wiki/File:Gatton_Railway_Bridge.JPG
  • 3. Forces in Pratt truss Image source :-http://www.slideshare.net/sheikhjunaidyawar/trusses-28955458 F
  • 5. Loading Train’s engine has highest weight as compared to other coaches Indian locomotive class WAP-5 • Length = 18.16 m • Weight = 79251.7 kg Source :- https://en.wikipedia.org/wiki/Indian_locomotive_class_WAP-5
  • 7. 18 m 7 m Dead Load = 150 kN/ meter/ side Total dead load = 150 x 18 = 2700 KN Live load = 800 KN 550 KN 550 KN 550 KN 550 KN 550 KN 1 2 3 4 5 6 7 12 11 10 9 8
  • 8. Objective Function:- min. total Area = min A(1)+A(2)+A(3)+A(4)+A(5)+A(6)+A(7)+A(8) +A(9)+A(10)+A(11)+A(12)+A(13)+A(14)+A(15)+ +A(16)+A(17)+A(18)+A(19)+A(20)+A(21)+A(22) Constraints :- min. Area = 0.0010 m^2 = 10 cm^2 disp at any node < 50 mm
  • 9. clc; clear all; % initial cromosome having area in the range 0.002 to 0.02 m^2- C = 2e-3*[1 2 4 10 3 9 10 7 1 5 1 1 2 4 10 3 9 10 7 1 5 2; 9 3 7 9 4 7 5 2 4 9 1 9 3 7 9 4 7 5 2 4 9 3; 7 3 4 1 1 7 3 2 4 9 8 7 3 4 1 1 7 3 2 4 9 4; 1 5 7 9 4 9 5 2 4 9 6 1 5 7 9 4 9 5 2 4 9 5; 4 10 6 2 2 2 3 2 4 1 9 4 10 6 2 2 2 3 2 4 1 6; 9 3 5 9 5 7 5 1 4 9 1 9 3 5 9 5 7 5 1 4 9 7; 9 9 4 1 4 3 1 2 4 9 8 9 9 4 1 4 3 1 2 4 9 8; 1 3 7 9 1 7 5 5 4 4 1 1 3 7 9 1 7 5 5 4 4 9; 9 8 3 1 2 4 2 2 4 9 6 9 8 3 1 2 4 2 2 4 9 10; 2 3 7 9 3 7 5 9 4 3 1 2 3 7 9 3 7 5 9 4 3 1; 8 7 2 10 4 5 3 10 4 7 4 8 7 2 10 4 5 3 10 4 7 2; 1 2 4 10 3 9 10 7 1 5 1 1 2 4 10 3 9 10 7 1 5 3; 9 3 7 9 4 7 5 2 4 9 1 9 3 7 9 4 7 5 2 4 9 4; 7 3 4 1 1 7 3 2 4 9 8 7 3 4 1 1 7 3 2 4 9 5; 1 5 7 9 4 9 5 2 4 9 6 1 5 7 9 4 9 5 2 4 9 6; 4 10 6 2 2 2 3 2 4 1 9 4 10 6 2 2 2 3 2 4 1 7; 9 3 5 9 5 7 5 1 4 9 1 9 3 5 9 5 7 5 1 4 9 8; 9 9 4 1 4 3 1 2 4 9 8 9 9 4 1 4 3 1 2 4 9 9; 1 3 7 9 1 7 5 5 4 4 1 1 3 7 9 1 7 5 5 4 4 10; 9 8 3 1 2 4 2 2 4 9 6 9 8 3 1 2 4 2 2 4 9 1; 2 3 7 9 3 7 5 9 4 3 1 2 3 7 9 3 7 5 9 4 3 2; 2 3 7 9 3 7 5 9 4 3 1 2 3 7 9 3 7 5 9 4 3 3];
  • 10. F_obj= sum(C,2); % return sum of rows in a column matrix of 22 X 1 %% selection Fitness = zeros(22,1); for a = 1:22 Fitness(a) = (1/(1+F_obj(a))); end S = sum(Fitness); Prob_of_cromosome = Fitness/S; roullet = zeros(22,1); roullet(1) = Prob_of_cromosome(1); for a = 2:22; roullet(a) = roullet(a-1) + Prob_of_cromosome(a); %CDF end
  • 11. %% new set of cromosome for a = 1 : 22 r = rand; if (r <= roullet(1)) new_cromosome_id = 1; else for b = 1 : 21 if (r > roullet(b) & r <= roullet(b+1)) new_cromosome_id = b+1; end end end C(a,:) = C (new_cromosome_id,:); end
  • 12. %% Cross Over C_new = C; for a= 1:22 r_crossover = rand(); if(r_crossover < 0.7) r_location = 10*round(rand(),1); % generates random number between 0 to 10. It selects location for crossover for b = r_location+1 : 22 if (a==22) C_new(a,b)= C(1,b); C_new(1,b) = C(22,b); else C_new(a,b)= C(a+1,b); C_new(a+1,b) = C(a,b); end end end end
  • 13. %% Mutation Probability 0.1 for a=1:22 for b=1:22 if(rand < 0.1) C_new(a,b)= round(((0.02-0.001)*rand+0.001),3); end end end
  • 14. %% Truss code coordinate=[0 0 ;3 0 ;6 0;9 0; 12 0; 15 0; 18 0;15 7; 12 7; 9 7; 6 7; 3 7;]; connectivity=[1 2;2 3;3 4;4 5;5 6; 6 7; 7 8;8 9; 9 10;10 11; 11 12;1 12; 2 12;3 12;3 11; 4 11;4 9; 4 10; 5 9; 5 8;6 8; 7 8]; boundary=[1 1 ; 0 0 ; 0 0 ;0 0 ;0 0;0 0; 1 1;0 0; 0 0;0 0;0 0;0 0]; load=[0 0 0 -550e3 0 -550e3 0 -550e3 0 -550e3 0 -550e3 0 0 0 0 0 0 0 0 0 0 0 0]; Elasticity=2.E+11*[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1];
  • 15. %% Penalty abs_unknown_displacement = abs(unknown_displacement); count = 0; for k = 1 : 22 for l = 1:20 if(abs_unknown_displacement(k,l)-0.050 > 0) count = count + 1; end C_new(k,:)= (1+0.05*count)*C_new(k,:); end end
  • 18. member 1 2 3 4 5 6 7 8 9 10 11 0.019 0.004 0.016 0.015 0.019 0.008 0.005 0.008 0.012 0.014 0.019 0.019 0.015 0.016 0.003 0.009 0.015 0.005 0.008 0.003 0.014 0.003 0.019 0.015 0.016 0.017 0.009 0.008 0.005 0.006 0.01 0.008 0.019 0.019 0.015 0.016 0.003 0.005 0.014 0.005 0.011 0.003 0.014 0.003 0.019 0.004 0.016 0.015 0.019 0.019 0.005 0.006 0.01 0.008 0.015 0.019 0.015 0.016 0.003 0.005 0.016 0.013 0.008 0.003 0.014 0.003 0.019 0.011 0.016 0.003 0.005 0.016 0.009 0.008 0.003 0.014 0.003 0.019 0.004 0.016 0.003 0.005 0.016 0.013 0.017 0.003 0.003 0.003 0.019 0.015 0.016 0.015 0.019 0.008 0.005 0.008 0.012 0.014 0.003 0.019 0.004 0.016 0.019 0.011 0.008 0.005 0.006 0.01 0.008 0.005 0.008 0.015 0.016 0.015 0.019 0.008 0.005 0.008 0.016 0.014 0.003 0.019 0.015 0.016 0.009 0.005 0.016 0.013 0.008 0.003 0.014 0.003 0.006 0.004 0.016 0.015 0.019 0.008 0.005 0.008 0.012 0.014 0.016 0.019 0.015 0.016 0.003 0.005 0.016 0.013 0.008 0.003 0.014 0.016 0.019 0.015 0.016 0.007 0.019 0.014 0.019 0.008 0.019 0.014 0.003 0.019 0.002 0.016 0.015 0.005 0.016 0.013 0.008 0.003 0.014 0.003 0.019 0.004 0.016 0.015 0.019 0.014 0.005 0.008 0.019 0.014 0.003 0.019 0.015 0.016 0.003 0.005 0.015 0.005 0.008 0.003 0.016 0.003 0.019 0.015 0.006 0.003 0.009 0.016 0.013 0.008 0.003 0.014 0.003 0.019 0.015 0.009 0.017 0.009 0.002 0.009 0.009 0.01 0.015 0.016 0.019 0.015 0.016 0.014 0.005 0.016 0.013 0.008 0.003 0.014 0.003 0.019 0.015 0.016 0.003 0.005 0.016 0.013 0.008 0.003 0.014 0.003 value 0.019 0.015 0.016 0.003 0.005 0.015 0.005 0.008 0.003 0.014 0.003 times 20 14 20 9 6 14 10 16 12 17 15
  • 19. 12 13 14 15 16 17 18 19 20 21 22 0.008 0.014 0.016 0.007 0.002 0.018 0.016 0.014 0.01 0.006 0.002 0.008 0.014 0.012 0.007 0.002 0.011 0.016 0.008 0.017 0.006 0.002 0.004 0.014 0.012 0.016 0.002 0.013 0.017 0.003 0.016 0.012 0.006 0.008 0.014 0.012 0.008 0.002 0.018 0.016 0.014 0.01 0.006 0.008 0.008 0.014 0.012 0.007 0.002 0.018 0.016 0.014 0.01 0.006 0.002 0.008 0.014 0.012 0.007 0.002 0.012 0.016 0.014 0.02 0.006 0.002 0.008 0.014 0.019 0.007 0.002 0.018 0.016 0.014 0.01 0.019 0.002 0.008 0.014 0.012 0.007 0.002 0.018 0.016 0.014 0.01 0.006 0.002 0.008 0.014 0.016 0.007 0.002 0.018 0.016 0.014 0.017 0.006 0.002 0.004 0.014 0.012 0.016 0.002 0.012 0.017 0.003 0.016 0.012 0.006 0.008 0.014 0.002 0.007 0.002 0.018 0.016 0.014 0.01 0.006 0.002 0.008 0.02 0.016 0.001 0.002 0.018 0.016 0.014 0.01 0.006 0.002 0.008 0.019 0.017 0.007 0.002 0.018 0.019 0.014 0.01 0.006 0.002 0.008 0.014 0.012 0.007 0.002 0.018 0.016 0.014 0.01 0.006 0.002 0.008 0.001 0.012 0.007 0.002 0.018 0.011 0.014 0.01 0.006 0.002 0.008 0.014 0.012 0.007 0.002 0.018 0.016 0.014 0.01 0.006 0.002 0.008 0.014 0.012 0.007 0.004 0.018 0.016 0.014 0.01 0.006 0.002 0.008 0.014 0.012 0.007 0.002 0.018 0.016 0.008 0.013 0.006 0.002 0.008 0.014 0.012 0.007 0.002 0.018 0.016 0.014 0.01 0.006 0.002 0.008 0.015 0.012 0.016 0.002 0.002 0.017 0.017 0.007 0.012 0.002 0.008 0.014 0.012 0.007 0.014 0.018 0.016 0.014 0.01 0.006 0.002 0.008 0.014 0.012 0.006 0.002 0.004 0.016 0.014 0.01 0.006 0.002 0.008 0.014 0.012 0.007 0.002 0.018 0.016 0.014 0.01 0.006 0.002 20 19 17 18 21 16 20 16 15 18 19