COMPLEX STRESS DISTRIBUTIONS IN ENGINEERING MATERIALS. 349 
where K,, K., K,, K, are arbitrary constants, and where 
P “Pp? We? P 
ar= - » +—— +——, 
4E2I?) gk 2K | (16) 
: Pp Wow? P | 
a + —_— ; 
s a/ dee gEL 2EL | 
The values of K,, K,, K,, K,—or rather, their relative magnitudes—ave determinable 
from the end conditions of the problem. But since they cannot all vanish 
simultaneously (for the shaft would then remain straight), we have only three 
ratios at our choice for satisfying four specified conditions ; hence, by elimination of 
the four constants, we obtain the criterion of neutral stability. 
It will be convenient in this section of the paper to take our origin at one end. 
Then, when the ends are simply supported we have the relations 
y =0,) 
and ay _M =O, i when s=0O or /, 
as* KI 
and substituting from (15) we find that 
Koa, = k=O) 
K, sin a/= 0. 
It follows that we must have s 
sin al= 0, = ‘ 2 : ; A GO) 
and, by (16), the criterion for neutral stability in this case is 
a 2 +4/ Pye 
2 2EI 4E22 gE 
BEY 5 Wa eh oA wile ft yh i er aie) 
mE) 7'gEl 
or 
When the ends are clamped, so that we have the relations 
y= 74 =0, when s=O or J, 
ds 
we find, on substitution from (15), that 
K,+K,=0, | 
aK,+6K,=0, !/ 
so that 
y=, (sin as - sinh Bs) + K, (cos as — cosh Bs). 
The criterion then takes the form 
sin al—* sinh Bl, cos al—cosh Bl, 
B co 
cos al —cosh Bl, — (sin al+— sinh ai), 
a 
and on expanding the determinant we obtain, finally, as the criterion for neutral 
stability, 
a? — B? 
2(1 —cos al cosh B21) = 
aand B being given by (16), so that l 
al? = 2[./B?+4C + Bl, | 
pl? = 2[,/B?+ 4C—B], 
where B and C are the quantities defined in equation (1). 
sin al sinh Bl, 
(19) 
