178 RECIPROCAL FIGURES, FRAMES, 



GENERAL THEORY OF DIAGRAMS OF STRESS IN THREE DIMENSIONS. 



./ Method of Representing Stress in a Body. 



DEFINITION*. A diagram of stress is a figure having such a relation to a 

 body under the action of internal forces, that if a surface A, limited by a 

 closed curve, is drawn in the body, and if the corresponding limited surface a 

 be drawn in the diagram of stress, then the resultant of the actual internal 

 forces on the positive side of the surface A in the body is equal and parallel 

 to the resultant of a uniform normal pressure p acting on the positive side of 

 the surface a in the diagram of stress. 



Let x, y, z be the co-ordinates of any point in the body, , 17, those of 

 the corresponding point in the diagram of stress, then 77, are functions of 

 x, y, z, the nature of which we have to ascertain, so that the internal forces 

 in the body may be in equilibrium. For the present we suppose no external 

 forces, such as gravity, to act on the particles of the body. We shall consider 

 such forces afterwards. 



THEOREM 1. If any closed surface is described in the body, and if the stress 

 on any element of that surface is equal and parallel to the pressure on the 

 corresponding element of surface in the diagram of stress, then the resultant stress 

 on the whole closed surface will vanish ; for the corresponding surface in the 

 diagram of stress is a closed surface, and the resultant of a uniform normal 

 pressure p on every element of a closed surface is zero by hydrostatics. 



It does not, however, follow that the portion of the body within the 

 closed surface is in equilibrium, for the stress on its surface may have a re- 

 sultant moment. 



THEOREM 2. To ensure equilibrium of every part of the body, it is neces- 

 sary and sufficient that 



f _dF _dF dF 



*~dx' ^~dy' 4 ~dz' 



where F is any function of x, y, and z. 



Let us consider the elementary area in the body dydz. The stress acting 

 on this area will be a force equal and parallel to the resultant of a pressure p 

 acting on the corresponding element of area in the diagram of stress. Resolving 



