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Project # 1
Problem 1-Design of Thin Cylindrical Vessel
Graphical solution
% ro=7850;
density, kg/m^3
% L=8;
length, m
% P=3.5e6
gas pressure, Pa
% E=210e9
stress, Pa
% R is the radius
% T is the thickness
% The problem can be written is standard optimization problem as follows:
%
Minimize
f(x1,x2) = 2*pi*ro*L*R*T+2*0.03*ro*pi*R^2
% subject to the following constraints:
%
Volum
g1(x1,x2) = pi*L*R^2 >= 25
%
Stress
g2(x1,x2) = P*(2-0.3)/(2*E)*R/T <= 0.001
%
Strain
g3(x1,x2) = P*R/T <= 210e6
%%
side
g4(x1,x2) = T <= 0
%%
side
g5(x1,x2) = R <= 0
%
%
%
%
WARNING : The hash marks for the inequality constraints must
%
be determined and drawn outside of the plot
%
generated by matlab
%
%---------------------------------------------------------------ro=7850; % kg/m^3
L=8;
%m
P=3.5e6 % gas pressure, Pa
E=210e9 % stress, Pa
x1=0:0.001:2;
x2=0:0.0005:0.1;
% x1 and x2 are vectors filled with numbers starting
% at 0 and ending at 2 and 0.1 with values at intervals of 0.001
[X1 X2] = meshgrid(x1,x2);
% generates matrices X1 and X2 correspondin
% vectors x1 and x2
% for clarity we define the foolowing variables
a1=2*pi*ro*L; a2=pi*L; a3=P*(2-0.3)/(2*E);
1
% objective function is given by
f1=a1.*X1.*X2+2*0.03*ro*pi.*X1.^2;
% The set inequality constraints are given by
ineq1=-a2.*X1.^2+25;
ineq2=a3.*X1./X2-0.001;
ineq3=3.5e6.*X1./X2-210e6;
ineq4=-X2;
ineq5=-X1;
% poltting the contours
[C1,h1] = contour(x1,x2,ineq1,[0,0],'r-');
set(h1,'LineWidth',1)
%clabel(C1,h1);
hold on
% allows multiple plots
k1 = gtext('g1=0');
% add text tothe figure
set(k1,'FontName','Times','FontWeight','bold','FontSize',10,'Color','red')
% will place the string 'g1' on the lot where mouse is clicked
[C2,h2] = contour(x1,x2,ineq2,[0,0],'r--');
%clabel(C2,h2);
set(h2,'LineWidth',1)
k2 = gtext('g2=0');
set(k2,'FontName','Times','FontWeight','bold','FontSize',10,'Color','red')
[C3,h3] = contour(x1,x2,ineq3,[0,0],'b-');
%clabel(C3,h3);
set(h3,'LineWidth',1)
k3 = gtext('g3=0');
set(k3,'FontName','Times','FontWeight','bold','FontSize',10,'Color','blue')
[C4,h4] = contour(x1,x2,ineq4,[0,0],'b--');
%clabel(C4,h4);
set(h4,'LineWidth',1)
k4 = gtext('g4=0');
set(k4,'FontName','Times','FontWeight','bold','FontSize',10,'Color','blue')
[C5,h5] = contour(x1,x2,ineq5,[0,0],'b--');
%clabel(C5,h5);
set(h5,'LineWidth',1)
2
k5 = gtext('g5=0');
set(k5,'FontName','Times','FontWeight','bold','FontSize',10,'Color','blue')
[C,h] = contour(x1,x2,f1,[8000,10000,20000],'g');
clabel(C,h);
set(h,'LineWidth',1)
% The equality and inequality constraints are not written
% with 0 on the right hand side. If you do write them that way
% you would have to include [0,0] in the contour commands
xlabel(' R (m)','FontName','times','FontSize',10,'FontWeight','bold');
% label for x-axes
ylabel(' t (m)','FontName','times','FontSize',10,'FontWeight','bold');
k6 = gtext('Project 1: Problem 1')
set(k6,'FontName','Times','FontSize',10,'FontWeight','bold')
%grid on
hold off
The result of graphical solution is obtained as follows:
3
% Problem 1 –Solution using MATLAB toolbox
clear
x0=[0.001 0.001];
xlb=[0.0001 0.0002 ];
% lower bound
xub=[] ;
% upper bound
options=optimset('LargeScale','off','Display','iter');
[x,f]=fmincon('objfunc1',x0,[],[],[],[],xlb,xub,'cons1',options)
function f = objfunc1(x)
a1=2*pi*7850*8; a2=pi*8; a3=3.5e6*(2-0.3)/(2*210e9);
f=a1.*x(1).*x(2)+2*0.03*7850*pi.*x(1).^2;
function [c, ceq] = cons1(x)
a1=2*pi*7850*8; a2=pi*8; a3=3.5e6*(2-0.3)/(2*210e9);
ineq1=-a2.*x(1).^2+25;
ineq2=a3.*x(1)./x(2)-0.001;
ineq3=3.5e6.*x(1)./x(2)-210e6;
ineq4=-x(2);
ineq5=-x(1);
c=[ineq1;ineq2;ineq3;ineq4;ineq5];
ceq=[];
The exact solution using MATLAB toolbox is obtained as:
R = 0.9974m , t = 0.0166 m , f = 8.0135e+003
4
% Project # 1
% Problem 2- Design of Flag Pole
% Graphical solution
% standard form of the optimization problem is :
% minimize f=(X1.^2-X2.^2);
% subject to
% g1=(a1+a2)./I-0.1;
% deflection
% g2=M*X1./(2*I)-165e6;
% bending stress
% g3=(a3./I).*(X2.^2+X2.*X1+X1.^2)-50e6; % shear stress
% g4=(X1+X2)./(X1-X2)-60;
% Mean diam./thickness
% g5=(X1-X2)/2-0.02;
% thickness< .02 m
% g6=-(X1-X2)/2+0.005;
% g7=X1-0.5;
% g8=0.05-X1;
% g9=X2-0.45;
% g10=0.04-X2;
% whre X1 is " do " and X2 is " di "
%---------------------------------------------------------------clear
x1=0:0.01:0.50;
% The semi-colon at the end prevents the echo
x2=0:0.01:0.50;
% These are also the side constraints
[X1 X2] = meshgrid(x1,x2);
% generates matrices X1 and X2 corresponding
% vectors x1 and x2
E=210e9;
ro=7800;
W=2000;
H=10;
P=4000;
M=(P*H+0.5*W*H^2);
S=(P+W*H);
A=(pi/4)*(X1.^2-X2.^2);
I=(pi/64)*(X1.^4-X2.^4);
a1=P*H^3/(3*E);
a2=W*H^4/(8*E);
a3=S/12;
%Pa
%kg/m^3
%N/m
%m
%N
%N.m
%N
%m^2
%m^4
f1=(X1.^2-X2.^2);
ineq1=(a1+a2)./I-0.1;
% deflection
ineq2=M*X1./(2*I)-165e6;
% bending stress
ineq3=(a3./I).*(X2.^2+X2.*X1+X1.^2)-50e6; % shear stress
ineq4=(X1+X2)./(X1-X2)-60;
% Mean diam./thickness
ineq5=(X1-X2)/2-0.02;
% thickness< .02 m
5
ineq6=-(X1-X2)/2+0.005;
ineq7=X1-0.5;
ineq8=0.05-X1;
ineq9=X2-0.45;
ineq10=0.04-X2;
[C1,h1] = contour(x1,x2,ineq1,[0,0],'r-');
% %clabel(C1,h1);
%
set(h1,'LineWidth',1)
% % ineq1 is plotted [at the contour value of 8]
%
hold on
% allows multiple plots
%
k1 = gtext('g1=0');
% add text tothe figure
set(k1,'FontName','Times','FontWeight','bold','FontSize',10,'Color','red')
%
% % will place the string 'g1' on the lot where mouse is clicked
%
[C2,h2] = contour(x1,x2,ineq2,[0,0],'r--');
% %clabel(C2,h2);
set(h2,'LineWidth',1)
k2 = gtext('g2=0');
set(k2,'FontName','Times','FontWeight','bold','FontSize',10,'Color','red')
[C3,h3] = contour(x1,x2,ineq3,[0,0],'b-');
% %clabel(C3,h3);
set(h3,'LineWidth',1)
k3 = gtext('g3=0');
set(k3,'FontName','Times','FontWeight','bold','FontSize',10,'Color','blue')
%will place the string 'g1' on the lot where mouse is clicked
[C4,h4] = contour(x1,x2,ineq4,[0,0],'b--');
%clabel(C4,h4);
hold on
set(h4,'LineWidth',1)
k4 = gtext('g4=0');
set(k4,'FontName','Times','FontWeight','bold','FontSize',10,'Color','blue')
[C5,h5] = contour(x1,x2,ineq5,[0,0],'b--');
%clabel(C5,h5);
set(h5,'LineWidth',1)
k5 = gtext('g5=0');
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set(k5,'FontName','Times','FontWeight','bold','FontSize',10,'Color','blue')
%
[C6,h6] = contour(x2,x1,ineq6,[0,0],'c--');
%clabel(C6,h6);
set(h6,'LineWidth',1)
k6 = gtext('g6=0');
set(k6,'FontName','Times','FontWeight','bold','FontSize',10,'Color','blue')
[C7,h7] = contour(x1,x2,ineq7,[0,0],'r--');
%clabel(C6,h6);
set(h7,'LineWidth',1)
k7 = gtext('g7=0');
set(k7,'FontName','Times','FontWeight','bold','FontSize',10,'Color','blue')
[C8,h8] = contour(x1,x2,ineq8,[0,0],'b--');
%clabel(C6,h6);
set(h8,'LineWidth',1)
k8 = gtext('g8=0');
set(k8,'FontName','Times','FontWeight','bold','FontSize',10,'Color','blue')
[C9,h9] = contour(x1,x2,ineq9,[0,0],'r--');
%clabel(C6,h6);
set(h9,'LineWidth',1)
k9 = gtext('g9=0');
set(k9,'FontName','Times','FontWeight','bold','FontSize',10,'Color','blue')
[C10,h10] = contour(x1,x2,ineq10,[0,0],'b--');
%clabel(C6,h6);
set(h10,'LineWidth',1)
k10 = gtext('g10=0');
set(k10,'FontName','Times','FontWeight','bold','FontSize',10,'Color','blue')
[C,h] = contour(x1,x2,f1,[0,0.0111,.1],'g');
clabel(C,h);
set(h,'LineWidth',1.5)
grid on
% The equality and inequality constraints are not written
% with 0 on the right hand side. If you do write them that way
% you would have to include [0,0] in the contour commands
xlabel(' do (m)','FontName','times','FontSize',12,'FontWeight','bold');
% label for x-axes
ylabel(' di (m)','FontName','times','FontSize',12,'FontWeight','bold');
k11 = gtext('project 1: problem 2')
set(k11,'FontName','Times','FontSize',12,'FontWeight','bold')
grid on
hold off
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The graphical solution is obtained as:
8
% Problem 2 –Solution using MATLAB toolbox
clear
x0=[.05 .6];
xlb=[0.05 0.04 ];
xub=[0.50 0.45] ;
% lower bound
% upper bound
options=optimset('LargeScale','off','Display','iter');
[x,f]=fmincon('objfunc2',x0,[],[],[],[],[],[],'cons2',options)
g=cons2(x)
function f = objfunc2(x)
f=(x(2)^2-x(1)^2);
function [c, ceq] = cons2(x)
E=210e9; %Pa
ro=7800; %kg/m^3
W=2000; %N/m
H=10; %m
P=4000; %N
M=(P*H+0.5*W*H^2); %N.m
S=(P+W*H);
%N
A=(pi/4)*(x(2).^2-x(1).^2); %m^2
I=(pi/64)*(x(2).^4-x(1).^4); %m^4
a1=P*H^3/(3*E);
a2=W*H^4/(8*E);
a3=S/12;
ineq1=(a1+a2)./I-0.1;
%ineq1=P*H^3./(3*E*I)+W*H^4./(8*E*I)-0.10;
ineq2=M*x(2)./(2*I)-165e6;
%ineq3=(S./(12*I)).*(x(2).^2+x(2).*x(1)+x(1).^2)-50e6;
ineq3=(a3./I).*(x(2).^2+x(2).*x(1)+x(1).^2)-50e6;
ineq4= (x(2)+x(1))/(x(2)-x(1))-60;
ineq5=(x(2)-x(1))/2-0.02;
ineq6=-(x(2)-x(1))/2+0.005;
ineq7=x(2)-0.5;
ineq8=0.05-x(2);
ineq9=x(1)-0.45;
ineq10=0.04-x(1);
% deflection
% bending stress
% shear stress
% Mean diam./thickness
% thickness< .02 m
c=[ineq1;ineq2;ineq3;ineq4;ineq5;ineq6;ineq7;ineq8;ineq9;ineq10];
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ceq = [];
The results obtained by MATLAB toolbox are
x = [ 0.4018
0.4154] and f = 0.0111
where x(1) is “ do” and x(2) is “ di”
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