574 hersey: stiffness of elastic systems 



The procedure for determining C is simple. Plot the values of 



S' 



log — — , observed in a series of model experiments, as ordinates 

 LE 



against a as abscissa; S' being any magnitude (for example, the 



weight needed to give a certain deflection of a spot of light on 



some arbitrary scale) which is proportional to the true stiffness, 



S. The value of C, at any part of this curve, will then be (1 + <x) 



times the slope of the curve. 



Note from (7) that, when C is positive, the /3 term has an 



opposite effect from the a term and may outweigh it. In fact, 



the condition for temperature compensation is 



I c > (10) 



= 1 + — approx. \ 



Numerical results in particular cases. For pure stretching or 

 bending, C = 0; for pure twisting or shearing, C = — 1. 



For a thin flat circular disc deflected at the center by a so- 

 called concentrated load, while freely supported at the rim, we 

 may take the deflection formula readily available in treatises on 

 elasticity, 6 and, by recasting it into the form of (3), just as if 

 it were the result of model experiments, and then applying (8), 

 obtain the expression 



c _ 2 (1 + <r) 2 



(l-<r) (3 + <r) . (11) 



= 1.5 for a = 0.3 



6 Thus from Love, Theory of Elasticity, 2nd edition, eq. (57J, p. 454, by put- 

 ting r = and h = and taking the value of D given by eq. (16). p. 443, we 

 find for the stiffness of an infinitely th : .n disc of radius a and thickness 2 h, 



a „ 32 ir /hy 



Comparing this expression with our eq. (3), and treating a as the linear dimen- 

 sion L, evidently 



const. 



<t> ( a ) = ; • 



(3+<r) (l-O 



Differentiating logarithmically, according to (8), immediately gives (11). In 



2«r 



the case of a disc clamped at the edge, the expression for C would be 



1 — a 



