In a similar manner,the stresses acting on the surface with the normal y are written in the x,y axes as: aiw- Therefore,the matrix of stresses in the x,y axes is: in setting: [P= c s -s c and observing that the matrix [P]is orthogonal ([p]=[P),one hast [oule.:[P][Guls>[P] where [P]is the transpose of matrix [P]. This expression can be developed to become 北 One can also rearrange the equation to be -2cs Ox 2cs 6y (11.4) Tet SC -SC (c2-s3J0 xy) Then: [o]=[T][o]y One has:Gsy=Pau 'P:Gn'P='Po Pou=dxP;au=PosP. 2003 by CRC Press LLCIn a similar manner, the stresses acting on the surface with the normal are written in the x,y axes as: Therefore, the matrix of stresses in the x,y axes is: in setting: and observing that the matrix [P] is orthogonal (t [P] = [P] –1), one has4 where t [P] is the transpose of matrix [P]. This expression can be developed to become One can also rearrange the equation to be (11.4) Then: 4 One has: sx,y = Ps
,t t P; s
t t P = t Psxy; Ps
,t = sx,yP; s
,t = t Psx,yP. y { } s/y x,y c s –s c sij [ ]
,t –s c Ó ˛ Ì ˝ Ï ¸ = { } sij x,y s/x,s/y [ ] c s –s c sij [ ]
,t c s – s c = = [ ] P c s –s c = sij [ ]
, t [ ] Pt sij [ ] = x y, [ ] P s
t
t t
t st c s – s c sx txy txy sy c s –s c = s
st Ót
t˛ Ô Ô Ì ˝ Ô Ô Ï ¸ c 2 s 2 2– cs s 2 c 2 2cs sc sc c2 s 2 – ( ) – sx sy Ótxy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ = [ ] s
,t = [ ] T [ ] s x,y TX846_Frame_C11 Page 225 Monday, November 18, 2002 12:26 PM © 2003 by CRC Press LLC