CHAPTER 13 COMPLEX STRESSES Summary The normal stress o and shear stress t on oblique planes resulting from direct loading are a=a sin20 and t=sin 20 The stresses on oblique planes owing to a complex stress system are: normal stress =(ox+)+(ax-)cos 20+txy sin 20 shear stress =(x-)sin 20-tx cos 20 The principal stresses (i.e.the maximum and minimum direct stresses)are then o1=(ox+0,)+√[(ax-,2+4t3] 02=(ox+oy)-√[(ox-0,)2+4t] and these occur on planes at an angle e to the plane on which ax acts,given by either tan28=,2” (ox-0) or tan0=p-0x where p=a1,or o2,the planes being termed principal planes.The principal planes are always at 90 to each other,and the planes of maximum shear are then located at 45 to them. The maximum shear stress is tmx=2√/[(ox-0y)2+4r,]=(o1-02) In problems where the principal stress in the third dimension a3 either is known or can be assumed to be zero,the true maximum shear stress is then (greatest principal stress-least principal stress) Normal stress on plane of maximum shear =(+ Shear stress on plane of maximum direct stress (principal plane)=0 Most problems can be solved graphically by Mohr's stress circle.All questions which are capable of solution by this method have been solved both analytically and graphically. 13.1.Stresses on oblique planes Consider the general case,shown in Fig.13.1,of a bar under direct load F giving rise to stress o,vertically. 326CHAPTER 13 COMPLEX STRESSES Summary The normal stress a and shear stress z on oblique planes resulting from direct loading are a = ay sin’ 8 and z = 30, sin 28 The stresses on oblique planes owing to a complex stress system are: normal stress = +(a, + ay) ++(ax - a,) cos 28 + zXy sin 28 shear stress = +(a, - cy) sin 28 - zXy cos 28 The principal stresses (i.e. the maximum and minimum direct stresses) are then a1 = *(a, + a,,) + ,J[ (a, -ay)’ + 45py] 0’ = *(a, + (ty) - $J[ (6, - o~)’ + 4~:,] and these occur on planes at an angle 0 to the plane on which a, acts, given by either where aP = alr or a’, the planes being termed principal planes. The principal planes are always at 90” to each other, and the planes of maximum shear are then located at 45” to them. The maximum shear stress is zmax= 3JC(~x-~,)2+4~~,l = a@, -a’) In problems where the principal stress in the third dimension u3 either is known or can be assumed to be zero, the true maximum shear stress is then +(greatest principal stress - least principal stress) Normal stress on plane of maximum shear = $(a, + a,) Shear stress on plane of maximum direct stress (principal plane) = 0 Most problems can be solved graphically by Mohr’s stress circle. All questions which are capable of solution by this method have been solved both analytically and graphically. 13.1. Stresses on oblique planes Consider the general case, shown in Fig. 13.1, of a bar under direct load F giving rise to stress by vertically. 326