This Matlab code simulates ship rolling behavior in irregular waves and calculates the probability of capsizing for different sea states. It first generates irregular wave time series data and calculates wave parameters. It then uses the Runge-Kutta method to solve the ship rolling differential equation to obtain roll angle time series. Probability distributions and CDFs are plotted for wave amplitude, heeling moment, roll angle and righting moment. The code also calculates the probability of various failure events and compares the capsizing probability over different sea states.

1. Matlab Code for Solution of the Equation of Ship Rolling, CDF & LRFD
close all
clear all
clc
sigmahat=0.5;
significant_waveheight=0.10;
g=9.8;
A=8.1e-3*g;
B=3.11/(significant_waveheight^2);
t_start(1)=0;
t_end=900;
t_step=0.01;
number_t_step=(t_end-t_start(1))/t_step;
wave_omega_start(1)=0.05;
wave_omega_end=6;
wave_omega_step=0.05;
number_wave_omega_step=(wave_omega_end-wave_omega_start(1))/wave_omega_step;
for j=1:number_t_step+1
summation_wave_amplitude(j)=0;
end
%for i=1:number_wave_omega_step+1
%phase(i)=6.2832*rand(1,1);% Phase angle is not included in the following
for loop because
% the phase angle will be changed in each time
step
%end
load ('random_phase_for_irregularwave.mat')
for j=1:number_t_step+1
for i=1:number_wave_omega_step+1 % this loop will sum the wave amplitude
at each time step over all frequencies
%S(i)=A*wave_omega_start(i)^-5*exp(-B*wave_omega_start(i)^-4);
S(i)=0.0081*((g^2)/(wave_omega_start(i)^5))*exp(-
0.032*(g/(significant_waveheight*(wave_omega_start(i)^2)))^2);
zeta(i)=sqrt(2*S(i)*wave_omega_step);
wave_amplitude=zeta(i)*cos(wave_omega_start(i)*t_start(j)+phase(i));
summation_wave_amplitude(j)=summation_wave_amplitude(j)+wave_amplitude;
if(i<number_wave_omega_step+1) %if statement is used to eleminate the
121th element of wave_omega_start
wave_omega_start(i+1)=wave_omega_start(i)+wave_omega_step;
end
end
if(j<number_t_step+1)%if statement is used to eleminate the 40002th
element of wave_omega_start
t_start(j+1)=t_start(j)+t_step;
end
end
figure
plot(t_start,summation_wave_amplitude)
title('Time vs Irregular Wave Amplitude');
xlabel('Time in second');
ylabel('Amplitude in meter');
figure
histfit(summation_wave_amplitude);

2. title('Histogram for Irregular Wave');
xlabel('Wave Amplitude');
ylabel('Frequency');
figure
[muhat_1,sigmahat_1] = normfit(summation_wave_amplitude);
pdfNormal_1 = normpdf(summation_wave_amplitude,muhat_1,sigmahat_1);
plot(summation_wave_amplitude, pdfNormal_1)
title('PDF for Irregular Wave');
xlabel('Wave Amplitude');
ylabel('f(Wave Amplitude)');
figure
cdfNormal_1=normcdf(summation_wave_amplitude,muhat_1, sigmahat_1);
plot(summation_wave_amplitude,cdfNormal_1)
title('CDF for Irregular Wave');
xlabel('Wave Amplitude');
ylabel('F(Wave Amplitude)');
time_period_roll=6.3366;
natural_omega=(2*3.1416)/time_period_roll;
density=997.04;
beam=[6.50 6.58 6.75 6.84 7.01 6.92 6.67 6.64 5.23 2.70];
draft=3.81;
displacement=227984;
metacentric_height=.947;
moment_of_inertia=(displacement*metacentric_height)/natural_omega^2;
damping_coefficient_1=0.01107;
damping_coefficient_2=0.01186;
%b_1=damping_coefficient_1/moment_of_inertia;
%b_2=damping_coefficient_2/moment_of_inertia;
b_1=0.14639;
b_2=0.14639;
for j=1:number_t_step+1% for each time step initially the summation of
heeling moment is considered 0
summation_heeling_moment(j)=0;
end
for i=1:number_wave_omega_step+1% This loop will calculte the coefficient of
right hand side/heeling moment at each frequency
time_period_wave(i)=(2*3.1416)/wave_omega_start(i);
wave_length(i)=(g/(2*3.1416))*time_period_wave(i)^2;
for j=1:number_t_step+1% this loop generates the wave amplitude at each
frequency over the defined time period.
wave_apmlitude(j)=zeta(i)*cos(wave_omega_start(i)*t_start(j)+phase(i));
end
wave_height(i)=max(wave_apmlitude)- min(wave_apmlitude);% wave height is
calculted for the wave of each frequency by subtracting minimum amplitude
from maximum amplitude
%reduction_coefficient_wave_slope(i)=atan(wave_height(i)/wave_length(i));
reduction_coefficient_wave_slope=0.682;
if (time_period_wave(i)>=11.2)
coeff_roll_add_mass=(((time_period_wave(i)-11.2)*.015)/2.7)+.07;
else
coeff_roll_add_mass=0.07-(((11.2-time_period_wave(i))*.015)/2.7);
end
%simpson rule
delata_l=1;
b44=(delata_l/3)*coeff_roll_add_mass*density*draft*beam.^3;

3. added_mass_moment(i)=b44(1)+b44(2)+b44(3)+b44(4)+b44(5)+b44(6)+b44(7)+b44(8)+
b44(9)+b44(10);
amplitude_e(i)=0.5*reduction_coefficient_wave_slope*(wave_height(i)/g)*wave_o
mega_start(i)^2*(abs(natural_omega^2-
(wave_omega_start(i)^2*added_mass_moment(i))/moment_of_inertia));
for j=1:number_t_step+1% this loop will sum the right hand side at each
frequency over the time period
heeling_moment=(1/(g*1000))*amplitude_e(i)*moment_of_inertia*cos(wave_omega_s
tart(i)*t_start(j)+phase(i));
summation_heeling_moment(j)=summation_heeling_moment(j)+heeling_moment;
end
end
figure
plot(t_start,summation_heeling_moment)
title('Time vs Heeling Moment');
xlabel('Time in second');
ylabel('Heeling Moment');
figure
histfit(summation_heeling_moment);
title('Histogram for Heeling Moment');
xlabel('Heeling Moment');
ylabel('Frequency');
figure
[muhat_2,sigmahat_2] = normfit(summation_heeling_moment);
muhat_2
sigmahat_2
pdfNormal_2 = normpdf(summation_heeling_moment,muhat_2,sigmahat_2);
plot(summation_heeling_moment, pdfNormal_2)
title('PDF for Heeling Moment');
xlabel('Heeling Moment');
ylabel('f(Heeling Moment)');
figure
cdfNormal_2=normcdf(summation_heeling_moment,muhat_2, sigmahat_2);
plot(summation_heeling_moment,cdfNormal_2)
title('CDF for Heeling Moment');
xlabel('Heeling Moment');
ylabel('F(Heeling Moment)');
f_1=inline('-b_1*v-b_2*v*abs(v)-natural_omega^2*x+summation_right_side');%
According to the runge kutta method the second order differential
%equation should be represented as :
%v_dot+b_1*v+b_2*v*abs(v)+natural_omega^2*x=summation_right_side
x_initial(1)=0;% initial roll angle is zero
v_initial(1)=0;% initial roll velocity is zero
for j=1:number_t_step+1% Right hand side is a function of time
summation_right_side_A(j)=0;%
summation_right_side_B(j)=0;
summation_right_side_C(j)=0;
end
% the following loop will sum the right hand side at first for each time
% step
for j=1:number_t_step+1

4. for i=1:number_wave_omega_step+1% this loop wiil sum the right hand side
for each time step over all frequencies
right_hand_side_A=amplitude_e(i)*cos(wave_omega_start(i)*t_start(j)+phase(i))
;% according to the runge kutta procedure only t_start(j) is used for right
hand side
summation_right_side_A(j)=summation_right_side_A(j)+right_hand_side_A;% right
hand side is summed for each time step which will be used in dv_1.
right_hand_side_B=amplitude_e(i)*cos(wave_omega_start(i)*(t_start(j)+t_step*0
.5)+phase(i));% according to the runge kutta procedure only
t_start(j)+t_step*0.5 is used for right hand side
summation_right_side_B(j)=summation_right_side_B(j)+right_hand_side_B;%right
hand side is summed for each time step which will be used in dv_2 & dv_3.
right_hand_side_C=amplitude_e(i)*cos(wave_omega_start(i)*(t_start(j)+t_step)+
phase(i));% according to the runge kutta procedure only t_start(j)+t_step is
used for right hand side
summation_right_side_C(j)=summation_right_side_C(j)+right_hand_side_C;%right
hand side is summed for each time step which will be used in dv_4.
end
end
for j=1:number_t_step+1
dx_1=t_step*v_initial(j);
dv_1=t_step*f_1(b_1,b_2,natural_omega,summation_right_side_A(j),v_initial(j),
x_initial(j));
dx_2=t_step*(v_initial(j)+(dv_1/2));
dv_2=t_step*f_1(b_1,b_2,natural_omega,summation_right_side_B(j),v_initial(j)+
0.5*dv_1,x_initial(j)+dx_1*0.5);
dx_3=t_step*(v_initial(j)+(dv_2/2));
dv_3=t_step*f_1(b_1,b_2,natural_omega,summation_right_side_B(j),v_initial(j)+
0.5*dv_2,x_initial(j)+dx_2*0.5);
dx_4=t_step*(v_initial(j)+(dv_3));
dv_4=t_step*f_1(b_1,b_2,natural_omega,summation_right_side_C(j),v_initial(j)+
dv_3,x_initial(j)+dx_3);
if(j<number_t_step+1)
x_initial(j+1)=x_initial(j)+(1/6)*(dx_1+2*dx_2+2*dx_3+dx_4);%
x_initial is the roll angle
v_initial(j+1)=v_initial(j)+(1/6)*(dv_1+2*dv_2+2*dv_3+dv_4);
end
end
for j=1:number_t_step+1
x_initial_1(j)=x_initial(j)*57.2975;
end
figure
plot(t_start,x_initial_1)
title('Time vs Roll Angle');
xlabel('Time in second');
ylabel('Roll Angle in Degree');
figure

5. histfit(x_initial_1);
title('Histogram for Roll Angle');
xlabel('Roll Angle');
ylabel('Frequency');
figure
[muhat_3,sigmahat_3] = normfit(x_initial_1);
pdfNormal_3 = normpdf(x_initial_1,muhat_3,sigmahat_3);
plot(x_initial_1, pdfNormal_3)
title('PDF for Roll Angle');
xlabel('Roll Angle');
ylabel('f(Roll Angle)');
figure
cdfNormal_3=normcdf(x_initial_1,muhat_3, sigmahat_3);
plot(x_initial_1,cdfNormal_3)
title('CDF for Roll Angle');
xlabel('Roll Angle');
ylabel('F(Roll Angle)');
for j=1:number_t_step+1
righting_moment(j)=(1/1000)*displacement*metacentric_height*sin(x_initial(j))
;
end
figure
plot(t_start,righting_moment)
title('Time vs Righting Moment');
xlabel('Time in second');
ylabel('Righting Moment');
figure
histfit(righting_moment);
title('Histogram for Righting Moment');
xlabel('Righting Moment');
ylabel('Frequency');
figure
[muhat_4,sigmahat_4] = normfit(righting_moment);
muhat_4
sigmahat_4
pdfNormal_4 = normpdf(righting_moment,muhat_4,sigmahat_4);
plot(righting_moment, pdfNormal_4)
title('PDF for Righting Moment');
xlabel('Righting Moment');
ylabel('f(Righting Moment)');
figure
cdfNormal_4=normcdf(righting_moment,muhat_4, sigmahat_4);
plot(righting_moment,cdfNormal_4)
title('CDF for Righting Moment');
xlabel('Righting Moment');
ylabel('F(Righting Moment)');
%probability calculation
A=find(abs(righting_moment)<abs(summation_heeling_moment));
cases_rm_less_then_hm=length(A)
total_number_of_events=number_t_step+1
P_A=cases_rm_less_then_hm/total_number_of_events% probability that righting
moment is less then heeling moment
deck_immersion_angle=4.89;
B=find(abs(x_initial_1)>deck_immersion_angle);
cases_ra_greater_then_deack_angle=length(B)
P_B=cases_ra_greater_then_deack_angle/total_number_of_events

6. cases_for_A_and_B=0;
for j=1:number_t_step+1
if((abs(x_initial_1(j))>deck_immersion_angle) &&
(abs(righting_moment(j))<abs(summation_heeling_moment(j))))
cases_for_A_and_B=cases_for_A_and_B+1;
end
end
cases_for_A_and_B
P_C=cases_for_A_and_B/total_number_of_events
p=1;
q=500;
r(1)=1;
for m=1:160
l=1;
for n=p:q
segment_righting_moment_(m,l)=abs(righting_moment(n));
segment_heeling_moment_(m,l)=abs(summation_heeling_moment(n));
l=l+1;
end
[segment_muhat_4(m),segment_sigmahat_4(m)]=normfit(segment_righting_moment_(m
,1:500));
[segment_muhat_2(m),segment_sigmahat_2(m)]=normfit(segment_heeling_moment_(m,
1:500));
z(m)=(segment_muhat_4(m)-
segment_muhat_2(m))/sqrt(segment_sigmahat_4(m)*segment_sigmahat_4(m)+segment_
sigmahat_4(m)*segment_sigmahat_4(m));
p=p+500;
q=q+500;
if(m<160)%if statement is used to eleminate the 40002th element of
wave_omega_start
r(m+1)=r(m)+1;
end
end
figure
plot(r,z)
title('"Z" values for the each segment from the total 800 second window');
xlabel('Segment No');
ylabel('"Z"');
p_normal=normcdf(z);
figure
plot(r,p_normal)
title('probability for the each segment from the total 800 second window');
xlabel('Segment No');
ylabel('probability');
p=1;
q=40;
r1(1)=1;
for m=1:4
average_p_normal(m)=0;
for n=p:q
average_p_normal(m)=average_p_normal(m)+p_normal(n);
end

7. mean_p_normal(m)=average_p_normal(m)/40;
p=p+40;
q=q+40;
if(m<4)%if statement is used to eleminate the 40002th element of
wave_omega_start
r1(m+1)=r1(m)+1;
end
end
pp = polyfit(r1,mean_p_normal,3);
x1 = linspace(1,4);
y1_2 = polyval(pp,x1);
figure
plot(r1,mean_p_normal,'o')
hold on
plot(x1,y1_2)
hold off
save('average_p_normal.mat','y1_2','-append')
Matlab Code for the Comparison of Capsizing Probability for Different Sea State
close all
clear all
clc
load('average_p_normal.mat')
figure
plot(x1,y1_1,'k')
hold on
plot(x1,y1_2,'c')
hold on
plot(x1,y1_3,'b')
hold on
plot(x1,y1_4,'m')
hold on
plot(x1,y1_5,'g')
hold on
plot(x1,y1_6,'y')
hold on
plot(x1,y1_7,'r')
hold on
ylabel('Probaility of Capsizing')
xlabel('segment no')
legend('Sea Stae 1','Sea state 2','Sea state 3','Sea state 4','Sea state
5','Sea state 6','Sea state 7')
box off
a2 = axes('XAxisLocation', 'Top');
set(a2, 'color', 'none')
set(a2, 'YTick', [])
set(a2, 'XLim', [0 800])
xlabel(a2,'Time')