# strs3d - 3D Stress Transformations

A Fortran code for three-dimensional stress transformations as outlined in Module 10 is listed below. A PC-executable version is also available, which can be saved to disk and run as a console program.

### Fortran Source

```c
c                     s t r s 3 d
c
c     an instructional code for transformations and principal values
c     of three-dimensional stress states
c
dimension sig(3,3),sigp(3),temp(3,3),signew(3,3),
1   t(3,3),tt(3,3)
common /inv/a(4)
c
c
10 write (6,15)
15 format (5x,'enter stress matrix by rows'/)
c
c
20 write (6,30)
30 format (/5x,'enter problem option:',
1        /7x,'1 - transformation',
2        /7x,'2 - principal values',
3        /7x,'3 - restart',
4        /7x,'4 - stop'/)
go to (100,200,10,35),ibr
c
35 stop 'program terminated'
100 continue
c
c**********************************************************************
c
c     compute transformed stresses
c
c**********************************************************************
c
write (6,120)
120 format (/5x,'enter euler angles psi, theta, phi (degrees):'/)
c
c     compute transformation matrix t(i,j)
c
c
t(1,1)=cpsi*cphi-sphi*ctheta*spsi
t(2,1)=-sphi*cpsi-spsi*ctheta*cphi
t(3,1)=stheta*spsi
t(1,2)=cphi*spsi+sphi*ctheta*cpsi
t(2,2)=-sphi*spsi+cphi*ctheta*cpsi
t(3,2)=-stheta*cphi
t(1,3)=stheta*sphi
t(2,3)=stheta*cphi
t(3,3)=ctheta
c
c     inverse (transpose) transformation matrix tt(i,j)
c
do 130 i=1,3
do 130 j=1,3
130 tt(i,j)=t(j,i)
c
c     compute transformed stresses signew = t*(sig*tt) = t*temp
c
call mult (sig,tt,3,3,3,temp)
call mult (t,temp,3,3,3,signew)
c
c     print results, then branch for new option
c
write (6,140)
140 format (//1x,'transformation matrix:',18x,'stress:'/)

do 150 i=1,3
150 write (6,160) (t(i,j),j=1,3),(signew(i,j),j=1,3)
160 format (1x,3f10.4,10x,3f10.4)
c
go to 20

c
c********************************************************************
c
c     compute principal values and directions of principal planes
c
c********************************************************************
c
c     compute invariants
c
200 a(1)=1.
a(2)=-1.*(sig(1,1)+sig(2,2)+sig(3,3))
a(3)= sig(1,1)*sig(2,2) + sig(1,1)*sig(3,3) + sig(2,2)*sig(3,3)
1     -sig(1,2)*sig(1,2) - sig(2,3)*sig(2,3) - sig(1,3)*sig(1,3)
a(4)=-1.*(sig(1,1)*(sig(2,2)*sig(3,3) - sig(3,2)*sig(2,3))
1         -sig(1,2)*(sig(2,1)*sig(3,3) - sig(3,1)*sig(2,3))
2         +sig(1,3)*(sig(2,1)*sig(3,2) - sig(3,1)*sig(2,2)))
c
c     compute principal stresses as roots of cubic equation
c
c
c     interactive evaluation of characteristic equation
c
205 write (6,210)
210 format (/5x,'enter solution option:',
1        /7x,'1 - evaluate characteristic equation',
2        /7x,'2 - newton-raphson interation',
3        /7x,'3 - compute direction cosines',
4        /7x,'4 - return for new problem option'/)
go to (215,220,245,20),ibr
c
c
215 write (6,216)
216 format (' Enter sigma (999 to stop)'/)
if (sigg.eq.999) go to 205
ysg=ysig(sigg)
write (6,217)  sigg,ysg
217 format (' y(',g12.4,') = ',g12.4/)
go to 215
c
c     Newton-Raphson iteration to refine roots
c
220 kmax=10
222 write (6,223)
223 format (' Enter initial guess (999 to stop)'/)
if (sigpi.eq.999) go to 205
do 230 k=1,kmax
ysg=ypsig(sigpi)
if (ysg.ne.0.) go to 226
write (6,225)
225          format (' Zero slope - try another guess')
go to 222
226       delta=ysig(sigpi)/ysg
sigpi=sigpi-delta
if (sigpi.ne.0) go to 228
if (delta.lt.0.01) go to 240
go to 230
228       if (abs(delta/sigpi).lt.0.01) go to 240
230    continue
write (6,*) ' No convergence for root'
go to 222
240 write (6,241) sigpi,k
241 format (' root =',g12.4,' (',i3,' iterations)')
go to 222
c
c
c     compute direction cosines and print results
c
245 write (6,*) ' Enter 3 principal stresses'
write (6,250)
250 format (//5x,'stress',12x,'l',9x,'m',9x,'n'/)
c
do 270 i=1,3
c
a2=(sig(2,2)-sigp(i))*(sig(3,3)-sigp(i))-(sig(2,3)*sig(2,3))
b= sig(1,2)*(sig(3,3)-sigp(i))-(sig(1,3)*sig(2,3))
c=(sig(1,2)*sig(2,3))-sig(1,3)*(sig(2,2)-sigp(i))
c
akk=sqrt (a2*a2 + b*b + c*c)
ak=0.
if (abs(akk).gt. 0.001) ak=1./akk
c
al=a2*ak
am=b*ak
an=c*ak
c
write (6,260) sigp(i),al,am,an
260    format (1x,f10.4,5x,3f10.4)
270 continue
c
go to 20
end
c
c
function ysig(sig)
common /inv/ a(4)
ysig=a(1)*sig**3+a(2)*sig*sig+a(3)*sig+a(4)
return
end
c
c
function ypsig(sig)
common /inv/ a(4)
ypsig=3.*a(1)*sig*sig+2.*a(2)*sig+a(3)
return
end
c
c
subroutine mult (a,b,l,m,n,c)
c
c     returns the matrix product c = a * b
c
dimension a(l,m),b(m,n),c(l,n)
c
do 20 i=1,l
do 20 j=1,n
cc=0.
do 10 k=1,m
cc=cc+a(i,k)*b(k,j)
10   continue
20 c(i,j)=cc
return
end

```