COMPOUND STRAINS AND STRESSES 157 
2? _ sine? 
“1 aban)” /2( cos $ + sin 5) 
But since . is a very small angle, cos $ may be taken=1, and 
ino _¢ +2 Jo(1+¢). Agia £1 
ain 5 = 5 Therefore ——< /2 1+5 . Again ple /2. 
sin = 
4 
Hence -= z= /2(1+$)- J2=% »/ 2, and f= f=. 
But it has already been shown that prs a(1 +0). 
Therefore a = Za +o) or E=2C(1+¢). 
If the faces AG and DH of a cube AH (Fig. 221) be subjected to a 
_ normal pressure of intensity p, the edge AD will be shortened, and the 
strain Eodised will be p/E. If, in 
addition, the faces AE and BH be i G 
_ subjected to a uniform normal pres- A mea 
sure of intensity p, the edge AD will ! 
be lengthened, and the strain pro- Se-4--- Jy 
duced in AD by the pressures on pee 
AE and BH will be op/E. D C 
In like manner, normal pressures Fira. 221. 
_ of intensity p on the faces AC and 
_ FH will lengthen the edge AD, and the strain in the direction AD, due 
to the pressures on AC and FH, will be op/E. 
When all the faces are subjected to normal pressure of intensity p, it 
biti) 
oLii}i|> 
ST LEAS 
rTtt 
: follows that the strain produced in the direction AD will be 
4 a! 
; eS they z( 1- 20). 
;: 
But the strain in the direction of each edge will be the same, and the volume 
strain will be three times the above linear strain (Art. 81), ae Mag, = 
strain = ma -2c). But volume strain= z Therefore E SP — 2c) = K 
or E=3K(1 - 20). 
: From the equations E= 2C(1+«) and E=3K (1 — 2c), the following 
relations are easily obtained :— 
9CK 3EK CE 
ee) Sah es K=90-3E° 
; _E-20_3K-E_3K-2C 
i hee Seat 6K  20+6K° 
> 
It is evident that if any two of the four quantities, E, C, K, and ¢, 
be found by experiment, the other two can be calculated. 
