Physics and Mathematics to Geology. 239 



might be expected to calculate the dimensions of each piece, 

 so that for the maximum load it is to be subjected to it shall 

 just not pass beyond the limit of perfect elasticity. There 

 are, however, in general agencies, such as wind pressure, 

 dynamical action of a moving load, &c, whose effects are not 

 very fully understood and whose magnitude cannot always 

 be foreseen. Thus it is the custom to allow a wide margin 

 for contingencies. Now the limit of perfect elasticity seems 

 the natural quantity to employ in allowing for this margin, 

 but the uncertainties attending its determination are such 

 that it is customary to employ the breaking-load instead. 

 The breaking-load for the particular kind of stress the member 

 in question is to be exposed to is divided by some number, 

 e. g. 4 or 5, called a factor of safety, and the dimensions of the 

 member are calculated so that its estimated load shall not 

 exceed the quotient of the breaking-load by the factor of 

 safety. The engineer varies the factor of safety according 

 to the nature of the load, and according to the confidence he 

 possesses in the uniformity of the material and in the com- 

 pleteness of his knowledge as to the vicissitudes the structure 

 is exposed to. It has thus come to pass that attention has been 

 largely directed to the breaking-loads, and theories have been 

 constructed which aim professedly at supplying a law for the 

 tendency to rupture, under the most general stress-systems 

 possible, of materials whose rupture-points have been found 

 under the ordinary simple stress-systems employed in 

 experiment. 



There are only two such theories of rupture for isotropic 

 materials that at present possess any general repute. To 

 understand them the reader requires to know that for any 

 stress-system there are at every point in an isotropic elastic 

 material three principal stresses along three mutually ortho- 

 gonal directions, and likewise three principal strains, whose 

 directions coincide with those of the principal stresses. If 

 an imaginary small cube of the material be taken at the point 

 considered with its faces perpendicular respectively to the 

 three principal stresses, then no tangential stresses act over 

 these faces. In a bar under a uniformly distributed longi- 

 tudinal stress L per unit of cross section, two of the principal 

 stresses are everywhere zero, and the third is parallel to 

 the axis and equals L. If E be Young's modulus, and rj 

 Poisson's ratio for the material, supposed isotropic and elastic, 

 the greatest principal strain is everwhere L/E and its direction 

 is parallel to the axis. The two remaining principal strains 

 are each — 77L/E, and they may be supposed to have for their 

 directions any two mutually perpendicular linos in the cross 

 section of the bar. 



