300 REPORTS ON THE STATE OF SCIENCE, ETC. 
stress, and / the longitudinal stress (tensile) in the shell at any point distant 
x measured along the axis from some fixed point 0. Let P be the internal 
pressure, 
Then the circumferential strain % = Basti where E is Young’s Modulus, and 
R E mE 
1 Poisson’s Ratio. 
m 
So that 
ee ema 1) al PO RG 
m 
R 
Consider the equilibrium of a longitudinal strip of the sheil subtending an 
angle 3 at the axis. This may be considered to act as a beam supported in 
some arbitrary manner at the points of constraint, and to sustain a load in the 
plane of bending which will be the resultant in that plane of the internal pressure 
and the circumferential stress. The internal pressure P will produce a uniform 
load intensity equal to PR &¢ per unit length, while the component, in the plane 
of bending, of the load due to the circumferential stress q will be equal to gto 
The component of the longitudinal stress will be negligible. The resultant 
intensity of load per unit length in the plane of bending is therefore 
w=PR3¢—gtdo 
The equation connecting load and displacement in a beam is 
(fa 
am =w 
where 8 is the flexural stiffness, which for a thin plate is equal to 
m? A [4 
m?—1 
2 tH 
ha, hor, 2 
m*—1 : 12 
2 3 diz 
P , Er” © pee %*—pRdo—did 
So that we have am ? oe Mii 
Inserting the value of gy from equation (1), we obtain the equation of equilibrium 
m ¢t diz, Kt ft 
———— =o 
m?—1 12 dat R® mR (2) 
If it be assumed that the longitudinal stress is that due to the internal pressure 
when the ends of the tube are closed, 
_ PR 
I~ 
me Es dee. at 
a ij s E en —2=P! : = . ; : 
and the equation becomes 21 18 dnt t Re P (3) 
where Te ee 2m—1 
2m 
d4z 
It reduces to the form + 4n4z—b 
dx' 
where ni=- 3 m1 
PR m? F 
12 m?—1 , z 
— z S12 
Et? =m? 
* This statement involves the assumption that the circumferential stress in — 
the tube has no effect on the flexural stiffness. A more rigorous, but less simple, M 
analysis leading to equation (2) is given by Nicolson, luc. cit., pp. 205-214. 
