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seepage_concrete_dam_with_sheetpile_isotropic_soil.m
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%=========================================================================
% MATLAB script file to draw the flownets of a concrete dam with vertical
% sheet pile using the conformal mapping technique
% For further details, go to the link below:
% https://github.com/SubhadipN/Conformal_Mapping#seepage-analysis
%=========================================================================
% Prepared by SUBHADIP NASKAR, RESEARCH SCHOLAR, IIT GUWAHATI
%=========================================================================
clear all; clc; warning('off','all')
%=========================================================================
% INPUTS::
a1 = 20;
a2 = 18;
a3 = 4.5;
b1 = 18;
b2 = 1.5;
h1 = -15;
h2 = -2;
s = 12;
k = 3.5*10^(-8); % Coefficient of permeability (m/s)
fl = 17; % Numbers of flow lines (odd number)
yf = 2.00; % Depth factor: controls the y-axis limit of the plot
xf = 2.75; % Width factor: controls the x-axis limit of the plot
%=========================================================================
% DERIVED PARAMETERS::
A = [-2*b1 0];B = [-b1 0];C = [0 0];D = [0 -s];F = [b1 0];
h = h1-h2;
%=========================================================================
% PLOT OF THE GIVEN EMBANKMENT DAM::
for i = 1:9
if i == 1
x1 = A(1);
y1 = A(2);
x2 = F(1);
y2 = F(2);
elseif i == 2
x1 = B(1);
y1 = B(2);
x2 = B(1);
y2 = a1;
elseif i == 3
x1 = B(1);
y1 = a1;
x2 = B(1)+b2;
y2 = a1;
elseif i == 4
x1 = B(1)+b2;
y1 = a1;
x2 = B(1)+b2;
y2 = a2;
elseif i == 5
x1 = B(1)+b2;
y1 = a2;
x2 = C(1);
y2 = a3;
elseif i == 6
x1 = C(1);
y1 = a3;
x2 = C(1);
y2 = C(2);
elseif i == 7
x1 = C(1);
y1 = C(2);
x2 = D(1);
y2 = D(2);
elseif i == 8
x1 = B(1);
y1 = -h1;
x2 = 1.3*B(1);
y2 = -h1;
else
x1 = C(1);
y1 = -h2;
x2 = -0.3*B(1);
y2 = -h2;
end
if x1 ~= x2
xx = linspace(x1,x2,200);
m = (y2-y1)/(x2-x1);
yy = m.*xx-m*x1+y1;
else
yy = linspace(y1,y2,200);
xx = x1*diag(eye(length(yy)))';
end
figure (1);
xlim([-xf*b1 0.8*xf*b1]);
ylim([-yf*a1 1.1*a1]);
set(gca,'FontSize',16);
hold on;
box on;
if i <= 6
if i == 1
leg1 = plot(xx,yy,'b','linewidth',1.5);
end
plot(xx,yy,'b','linewidth',1.5)
elseif i == 7
leg2 = plot(xx,yy,'k','linewidth',4);
else
if i == 8
leg3 = plot(xx,yy,'b-.','linewidth',1.5);
end
plot(xx,yy,'b-.','linewidth',1.5)
end
end
%=========================================================================
% FLOW LINES IN W-PLANE::
f = 0.4;
ls = 100;
x = linspace(-k*h,0,fl);
y = linspace(0,-f*k*h,fl);
for ii = 1:fl
% '||' to x-axis
xx = linspace(0,-k*h,ls);
yx = y(ii)*diag(eye(ls));
wx(ii,1:ls) = complex(xx,yx');
% '||' to y-axis
xy = x(ii)*diag(eye(ls));
yy = linspace(0,f*k*h,ls);
wy(ii,1:ls) = complex(xy',yy);
end
%=========================================================================
% FLOW LINES IN Z-PLANE::
tb = -sqrt(b1^2+s^2)/s;
tx = 0.5*(1-tb)*cos(pi.*wx/(k*h))+0.5*(1+tb);
ty = 0.5*(1-tb)*cos(pi.*wy/(k*h))+0.5*(1+tb);
zx = -s*sqrt(tx.^2-1);
zy = s*sqrt(ty.^2-1);
zx(1,:) = complex(linspace(0,B(1),ls),0);
zy(1,:) = complex(linspace(B(1),A(1),ls),0);
sz = size(zx);
for jj = 2:sz(1)
for kk = 1:sz(2)
if imag(zx(jj,kk)) >= 0
zx(jj,kk) = -complex(real(zx(jj,kk)),imag(zx(jj,kk)));
end
if imag(zy(jj,kk)) >= 0
zy(jj,kk) = -complex(real(zy(jj,kk)),imag(zy(jj,kk)));
end
end
end
for jj = 2:sz(1)
for kk = 1:sz(2)
rzy = real(zy(jj,kk));
izy = imag(zy(jj,kk));
zy(jj,kk) = complex(-rzy,izy);
end
end
for ii = 1:fl
leg4 = plot(real(zx(ii,:)),imag(zx(ii,:)),'k--','linewidth',1.5);
end
for ii = 1:fl
leg5 = plot(real(zy(ii,:)),imag(zy(ii,:)),'r','linewidth',1.5);
end
xlabel('Width (m)','fontsize',16);
ylabel('Height (m)','fontsize',16)
legend([leg1 leg2 leg3 leg4 leg5],{'Dam outline','Sheet pile',...
'Water level','Flow lines','Equipotential lines'},...
'Location','northeast','fontsize',14);
for ii = 1:2:fl
t1 = ['\phi_{',num2str(ii),'} = ',num2str(x(ii))];
t2 = ['\psi_{',num2str(ii),'} = ',num2str(y(ii))];
if ii <= (fl+1)/2
xc1 = min(real(zy(ii,:)));
yc1 = min(imag(zy(ii,:)));
text(xc1,yc1,t1,'HorizontalAlignment','right',...
'VerticalAlignment','top','Color','r','fontsize',16)
else
xc1 = max(real(zy(ii,:)));
yc1 = min(imag(zy(ii,:)));
text(xc1,yc1,t1,'HorizontalAlignment','left',...
'VerticalAlignment','top','Color','r','fontsize',16)
end
if ii == 1
text(0.5*B(1),0,t2,'HorizontalAlignment','left',...
'VerticalAlignment','bottom','Color','b','fontsize',16)
elseif mod((ii+1)/2,2) == 0
xc2 = max(real(zx(ii,:)));
h = text(xc2,0.5,t2,'HorizontalAlignment','left','Color','k',...
'FontSize',16);
set(h,'Rotation',90);
elseif mod((ii+1)/2,2) == 1
xc2 = min(real(zx(ii,:)));
h = text(xc2,0.5,t2,'HorizontalAlignment','left','Color','k',...
'fontsize',16);
set(h,'Rotation',90);
end
end
t3 = ['z = s\surd {(t^2-1)}; '];
t4 = ['t = 0.5(1-t_B)cos(\piw/kh)+0.5(1+t_B); '];
t5 = ['t_B = -\surd ((b/s)^2+1)'];
text(A(1),1.2*min(min(imag(zy))),[t3 t4 t5],'HorizontalAlignment',...
'left','Color','k','fontsize',18);
axis equal; grid on; grid minor;
%=========================================================================