158-160] General Theory of Stress 143 



160. Assuming that equation (80) and the two similar equations are 

 satisfied, the normal component of stress across the plane of which the 

 direction cosines are I, m, n is 



IF + mG + nH = PP XX + m?P yy + ri*P zz + 2mn P yz +ZnlP zx 

 The quadric 



^l ...... (82) 



is called the stress-quadric. If r is the length of its radius vector drawn in 

 the direction I, m, n, we have 



r 2 (l*P xx + m*P yy + n*P zz + 2mnP yz + 2nlP zx + 2lmP xy ) = 1. 



It is now clear that the normal stress across any plane I, m, n is measured 

 by the reciprocal of the square of the radius vector of which the direction 

 cosines are I, m, n. Moreover the direction of the stress across any plane 

 /, m, n is that of the normal to the stress-quadric at the extremity of this 

 radius vector. For r being the length of this radius vector, the coordinates 

 of its extremity will be rl, rm, rn. The direction cosines of the normal at 

 this point are in the ratio 



rlP xx + rmP xy + rnP zx : rlP xy + rml^ y + rnP yz : rlP zx + rm P yz + rn P zz 

 or F : G : H, which proves the result. 



The stress-quadric has three principal axes, and the directions of these 

 are spoken of as the axes of the stress. Thus the stress at any point has 

 three axes, and these are always at right angles to one another. If a small 

 area be taken perpendicular to a stress axis at any point, the stress across 

 this area will be normal to the area. If the amounts of these stresses are 

 7?, J, J, then the equation of the stress-quadric referred to its principal 

 axes will be 



Clearly a positive principal stress is a simple tension, and a negative 

 principal stress is a simple pressure. 



As simple illustrations of this theory, it may be noticed that 



(i) For a simple hydrostatic pressure P, the stress-quadric becomes an imaginary 

 sphere 



The pressure is the same in all directions, and the pressure across any plane is at right 

 angles to the plane (for the tangent plane to a sphere is at right angles to the radius 

 vector). 



(ii) For a simple pull, as in a rope, the stress-quadric degenerates into two parallel 

 planes 



