hersey: stiffness of elastic systems 571 



such a way that the springs fail to meet by about an inch. 

 Holding the frame vertical, hang a weight to the free end of the 

 upper spring, and note the deflection. Then couple the two 

 springs together. The same weight will now cause but half 

 that deflection. The experiment can readily be extended to show 

 the effect of using springs of unequal stiffnesses, or of weights 

 hung not at the coupling. 



Influence of temperature and elastic after-effect on coupled sys- 

 tems. 5 Differentiating (1) gives for the fractional change in the 

 stiffness of the coupled system, in terms of the fractional changes 

 in the stiffnesses of the components, due to any cause whatever, 



(2) 

 , and 



S a ~\~ Sb 



depends evidently on the two components but not on X, and, 

 therefore, not on the manner of applying the external force. 

 The changes ds a and ds b may equally well be interpreted as 

 temperature effects, or elastic after-effects. In either case (2) 

 shows that the relative contributions of the two components are 

 fixed by the single factor r\ and are proportional to the respec- 

 tive stiffnesses. We therefore pass to the consideration of the 

 stiffness of a single body. 



General expression for the stiffness of a body. The treatment 

 of non-homogeneous or anisotropic bodies will be simplified by 

 the conception of generalized shape. Two bodies may be said 

 to have the same generalized shape when not simply linear 

 magnitudes, but all physical quantities associated with points 

 in the bodies, are similarly distributed in the two bodies; that 

 is, distributed so that the ratio of the two magnitudes of any 

 one such quantity is the same at all pairs of corresponding 

 points. Thus two pieces of rolled sheet metal of different sizes 

 and materials have the same generalized shape if they have been 

 cut out to the same geometrical shape, similarly oriented with 

 respect to the direction of rolling, and if the rolling has pro- 



5 The displacement due to temperature or to elastic after-effect at constant 

 load is assumed so small that, in differentiating (1), \ may be regarded as 

 constant. 



