§8.3 Introduction to Advanced Elasticity Theory 223 the X direction due to o is given by oa△S cosnx Stress components in the other axial directions will be similar in form. Thus,for equilibrium of forces in the X direction, PiAS+F,AS=u AS cosnx+tyAS cosny+iAS cosnz As h0(i.e.plane ABC passes through o),the second term above becomes very small and can be neglected.The above equation then reduces to Pxn =Oxx cos nx txy cos ny txz cos nz (8.1) Similarly,for equilibrium of forces in the y and z directions, Pyn =oyy cos ny +tyx cos nx tyz cos nz (8.2) Pan =Ou cos nz +tur cos nx tay cos ny (8.3) The resultant stress pn on the plane ABC is then given by pm=Vp层n+p品+p) (8.4) The normal stress on is given by resolution perpendicular to the face ABC, i.e. On Pxn cos nx Pyn cos ny Pin cos nz (8.5) and,by Pythagoras'theorem (Fig.8.4),the shear stress tn is given by tn =V(pi-2) (8.6) Ammmmro n C Fig.8.4.Normal,shear and resultant stresses on the plane ABC. It is often convenient and quicker to define the line of action of the resultant stress p by the direction cosines I'=cos(Pnx)=Pxn/Pn (8.7) m'=cos(Pny)=Pyn/pn (8.8) n'=cos(Pnz)=Pin/Pn (8.9)58.3 Introduction to Advanced Elasticity Theory 223 the X direction due to a, is given by a,AS cos nx Stress components in the other axial directions will be similar in form. Thus, for equilibrium of forces in the X direction, h 3 pxnAS + F,AS- = cxxAScosnx + t,!,AScosny + t,,AScosnz As h + 0 (i.e. plane ABC passes through Q), the second term above becomes very small and can be neglected. The above equation then reduces to pxn = a, COS nx + txy COS ny + r,, COS nz (8.1) Similarly, for equilibrium of forces in the y and z directions, (8.2) i (8.3) pYn = uyy cos ny + tyx cos nx + tyz cos nz pm = a, cos nz + tu cos nx + tzy cos ny The resultant stress pn on the plane ABC is then given by Pn Jb:n + P;n + Pzn) (8.4) The normal stress a, is given by resolution perpendicular to the face ABC, i.e. and, by Pythagoras’ theorem (Fig. 8.4), the shear stress sn is given by = pxn COS nx + Pyn COS ny + Pzn COS nz rn = J(Pi-4) Fig. 8.4. Normal, shear and resultant stresses on the plane ABC. It is often convenient and quicker to define the line of action of the resu-.ant stress pn by the direction cosines 1’ = cos(pnx) = PxnIpn (8.7)