%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%pingetensor
syms x y z
S=[3*x*y 5*y^2 0; 5*y^2 0 2*z; 0 2*z 0]
S =
[ 3*x*y, 5*y^2,     0]
[ 5*y^2,     0,   2*z]
[     0,   2*z,     0]

%silinder y^2+z^2-4=0
phi= y^2+z^2-4
phi =
y^2+z^2-4

% SILINDRI KÜLGPIND
% pinnanormaal:
nu=[diff(phi,x),diff(phi,y),diff(phi,z)]
nu =
[   0, 2*y, 2*z]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Punkt P
x=2,y=1,z=sqrt(3)
x =
     2
y =
     1
z =
    1.7321
% pingetensor
SP=subs(S), 
SP =
    6.0000    5.0000         0
    5.0000         0    3.4641
         0    3.4641         0
% pinnanormaal
nuP=subs(nu),
nuP =
         0    2.0000    3.4641
% ühiknormaal
nuP=nuP/norm(nuP)
nuP =
         0    0.5000    0.8660
% pingevektor
p=(nuP*SP)
p =
    2.5000    3.0000    1.7321
% pingevektori moodul
pm=norm(p)
pm =
    4.2720
% pingevektori proj. normaalil
sigma=p*nuP'
sigma =
    3.0000
% nihkepinge moodul
tau = sqrt(pm^2-sigma^2)
tau =
    3.0414
% nihkepinge sihiline ühikvektor
nuPp=cross(nuP,p);
tv=cross(nuPp,nuP)/norm(nuPp)
tv =
    0.8220    0.4932   -0.2847
% nihkepinge vektor
tauv=tau*tv
tauv =
    2.5000    1.5000   -0.8660

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Punkt Q
x=0,y=1,z=sqrt(3)
x =
     0
y =
     1
z =
    1.7321
% pingetensor
SP=subs(S), 
SP =
         0    5.0000         0
    5.0000         0    3.4641
         0    3.4641         0
% pinnanormaal
nuP=subs(nu),
nuP =
         0    2.0000    3.4641
% ühiknormaal
nuP=nuP/norm(nuP)
nuP =
         0    0.5000    0.8660
% pingevektor
p=(nuP*SP)
p =
    2.5000    3.0000    1.7321
% pingevektori moodul
pm=norm(p)
pm =
    4.2720
% pingevektori proj. normaalil
sigma=p*nuP'
sigma =
    3.0000
% nihkepinge moodul
tau = sqrt(pm^2-sigma^2)
tau =
    3.0414
% nihkepinge sihiline ühikvektor
nuPp=cross(nuP,p);
tv=cross(nuPp,nuP)/norm(nuPp)
tv =
    0.8220    0.4932   -0.2847
% nihkepinge vektor
tauv=tau*tv
tauv =
    2.5000    1.5000   -0.8660

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Punkt R
x=4,y=1,z=sqrt(3)
x =
     4
y =
     1
z =
    1.7321
% pingetensor
SP=subs(S), 
SP =
   12.0000    5.0000         0
    5.0000         0    3.4641
         0    3.4641         0
% pinnanormaal
nuP=subs(nu),
nuP =
         0    2.0000    3.4641
% ühiknormaal
nuP=nuP/norm(nuP)
nuP =
         0    0.5000    0.8660
% pingevektor
p=(nuP*SP)
p =
    2.5000    3.0000    1.7321
% pingevektori moodul
pm=norm(p)
pm =
    4.2720
% pingevektori proj. normaalil
sigma=p*nuP'
sigma =
    3.0000
% nihkepinge moodul
tau = sqrt(pm^2-sigma^2)
tau =
    3.0414
% nihkepinge sihiline ühikvektor
nuPp=cross(nuP,p);
tv=cross(nuPp,nuP)/norm(nuPp)
tv =
    0.8220    0.4932   -0.2847
% nihkepinge vektor
tauv=tau*tv
tauv =
    2.5000    1.5000   -0.8660

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Punkt L
x=2,y=2,z=0
x =
     2
y =
     2
z =
     0
% pingetensor
SP=subs(S), 
SP =
    12    20     0
    20     0     0
     0     0     0
% pinnanormaal
nuP=subs(nu),
nuP =
     0     4     0
% ühiknormaal
nuP=nuP/norm(nuP)
nuP =
     0     1     0
% pingevektor
p=(nuP*SP)
p =
    20     0     0
% pingevektori moodul
pm=norm(p)
pm =
    20
% pingevektori proj. normaalil
sigma=p*nuP'
sigma =
     0
% nihkepinge moodul
tau = sqrt(pm^2-sigma^2)
tau =
    20
% nihkepinge sihiline ühikvektor
nuPp=cross(nuP,p);
tv=cross(nuPp,nuP)/norm(nuPp)
tv =
     1     0     0
% nihkepinge vektor
tauv=tau*tv
tauv =
    20     0     0

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Punkt M
x=2,y=0,z=2
x =
     2
y =
     0
z =
     2
% pingetensor
SP=subs(S), 
SP =
     0     0     0
     0     0     4
     0     4     0
% pinnanormaal
nuP=subs(nu),
nuP =
     0     0     4
% ühiknormaal
nuP=nuP/norm(nuP)
nuP =
     0     0     1
% pingevektor
p=(nuP*SP)
p =
     0     4     0
% pingevektori moodul
pm=norm(p)
pm =
     4
% pingevektori proj. normaalil
sigma=p*nuP'
sigma =
     0
% nihkepinge moodul
tau = sqrt(pm^2-sigma^2)
tau =
     4
% nihkepinge sihiline ühikvektor
nuPp=cross(nuP,p);
tv=cross(nuPp,nuP)/norm(nuPp)
tv =
     0     1     0
% nihkepinge vektor
tauv=tau*tv
tauv =
     0     4     0

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% OTSPIND x=0
% otspinna ühiknormaal
nuP=[-1,0,0]
nuP =
    -1     0     0
% pinge melevaldses punktis
pmp=(nuP*S).'
pmp =
[ -3*x*y]
[ -5*y^2]
[      0]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Punkt Q
x=0,y=1,z=sqrt(3)
x =
     0
y =
     1
z =
    1.7321
% pingetensor
SP=subs(S), 
SP =
         0    5.0000         0
    5.0000         0    3.4641
         0    3.4641         0
% pingevektor
p=(nuP*SP)
p =
     0    -5     0
% pingevektori moodul
pm=norm(p)
pm =
     5
% pingevektori proj. normaalil
sigma=p*nuP'
sigma =
     0
% nihkepinge moodul
tau = sqrt(pm^2-sigma^2)
tau =
     5
% nihkepinge sihiline ühikvektor
nuPp=cross(nuP,p);
tv=cross(nuPp,nuP)/norm(nuPp)
tv =
     0    -1     0

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% OTSPIND x=4
% otspinna ühiknormaal
nuP=[1,0,0]
nuP =
     1     0     0
% pinge melevaldses punktis
pmp=(nuP*S).'
pmp =
[ 3*x*y]
[ 5*y^2]
[     0]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Punkt P
x=4,y=1,z=sqrt(3)
x =
     4
y =
     1
z =
    1.7321
% pingetensor
SP=subs(S), 
SP =
   12.0000    5.0000         0
    5.0000         0    3.4641
         0    3.4641         0
% pingevektor
p=(nuP*SP)
p =
    12     5     0
% pingevektori moodul
pm=norm(p)
pm =
    13
% pingevektori proj. normaalil
sigma=p*nuP'
sigma =
    12
% nihkepinge moodul
tau = sqrt(pm^2-sigma^2)
tau =
     5
% nihkepinge sihiline ühikvektor
nuPp=cross(nuP,p);
tv=cross(nuPp,nuP)/norm(nuPp)
tv =
     0     1     0

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Pinge silindrilise pinna meelevaldses punktis
syms x y z
% 1. sil. pinna normaal
nu
nu =
[   0, 2*y, 2*z]
% 2. sil. pinna ühiknormaal
nuno=sqrt(4*y^2+4*z^2)
nuno =
2*(y^2+z^2)^(1/2)
nuy=nu/nuno
nuy =
[                 0, y/(y^2+z^2)^(1/2), z/(y^2+z^2)^(1/2)]
% pinge meelevaldses punktis
pmp=((nuy*S).')
pmp =
[ 5*y^3/(y^2+z^2)^(1/2)]
[ 2*z^2/(y^2+z^2)^(1/2)]
[ 2*y/(y^2+z^2)^(1/2)*z]

echo off