ANALYSIS OF THRUST. 



57 



In the case in hand, then, one may consider the effect of any portion of 

 the forces shown in figure 26 by itself Take, first, six forces, one on each 

 face of the cube, all directed inwards and each equal to P/3. This group is 

 shown in figure oa, and, since each of the forces is supposed to be uniformly 

 distributed over the surface, it is manifest that their effect will be to com- 

 press the mass without any alteration of shape. In other words, this group 

 of forces represents a simple compressive stress, unaccompanied by distortion 

 of figure. 



There now remain eight forces, which may be separated into two groups 

 of four. One of these acts in a plane parallel to o x y, as show^n in figure 



Figure 3— Analysis of a Thrust. 



Sb, and the other in a plane parallel to o y z, as shown in figure 3c. Thus 

 the three diagrams of figure 3 show, in a segregated form, all the forces of 

 figure 2. 



The last two diagrams each show a pressure in one direction accompanied 

 by an exactly equal traction acting at right angles to it. Now it was shown 

 above that a 2)ressure produces a cubical compression and, since traction is 

 negative compression, a traction must produce cubical dilatation. Further- 

 more the compression produced by a given pressure is exactly equal to the 

 dilatation produced by a traction of equal intensity. Hence no change of 

 volume can be produced by the system of forces shown in figure Sb or 3c, 

 and the effect of the system of forces, or the strain, must consist in a simple 

 change of shape, just as if the mass ivere incompressible. 



*If the pressure of figure 36 were to act alone upon an incompressible cube 

 of unit volume, it would diminish its height by an amount proportional to 

 the pressure, according to the experimental result embodied in Hooke's law, 

 " Ut teusio, sic vis," or strain is proportional to stress. Hence if n is a cer^ 

 tain constant, called the modulus of rigidity, the force P/3 would shorten 

 the unit cube by an amount Pl'Sn. This contraction would be accompanied 

 by an increase of the other dimensions of the mass, since the volume re- 

 mains constant. Thus o x and o y would each be increased by a small 

 quantity, say d, and the value of d is to be found from the equation — 

 13 = (1 _ p/3>0 (1 + ^) (1 + d). 



IX— Bull. Geol. Soc. Am., Vol. 2, 1890. 



