330 Mechanics of Materials §13.4 y Fig.13.5.Two-dimensional complex stress system. The diagram thus represents a complete stress system for any condition of applied load in two dimensions and represents an addition of the stress systems previously considered in §§13.2and13.3. The formulae obtained in these sections may therefore be combined to give o=(ax+a)+(ax-a)cos 20+ixy sin 20 (13.8) and to=(ox-a)sin 20-txy cos 20 (13.9) The maximum and minimum stresses which occur on any plane in the material can now be determined as follows: For oe to be a maximum or minimum doe=0 de Now o=(ox+)+(ax-G)cos 20+txy sin 20 do =-(-,sin 20+2tx cos20-0 do or tan 20 Lisy (13.10) (ax-0,) from Fig.13.6 sin 20 = 2txy √/[(ox-o,)2+4t,] (x-0) cos 20=- √/[ax-,)2+4t] (G,-0+4Ty 2y 28 (-cy} Fig.13.6,330 Mechanics of Materials $1 3.4 t Fig. 13.5. Two-dimensional complex stress system The diagram thus represents a complete stress system for any condition of applied load in two dimensions and represents an addition of the stress systems previously considered in Gg13.2 and 13.3. The formulae obtained in these sections may therefore be combined to give ug = +(u, + u,) ++(u, - u,) cos 28 +7,, sin 28 (1 3.8) and zg= ~(u,-u,)sin28--z,,cos28 (1 3.9) determined as follows: The maximum and minimum stresses which occur on any plane in the material can now be For ag to be a maximum or minimum Now ~ =o 40 dg = +(a, + a,) ++(a, - a,) COS 20 + T,~ sin 20 3 = - (a, - ay) sin 20 + 25,, cos 20 = o de .. or .'. from Fig. 13.6 sin20 = - JW, - + 4e,1 (1 3.10) (u,-u,) Fig. 13.6