346 REPORTS’ON'THE STATE OFJSCIENCE, ETC. 
even when no difficulties of welding would be encountered. It also leads to some 
unexpected conclusions regarding the relation between the spacing of the rings and 
their effectiveness. 
It was shown in the previous paper that the radial displacement at any point of 
the pipe wall is given by the equation 
Bim? ® die, Bee 
m?—1° 12° dxt" R27’ 
where R is the mean radius of pipe, ¢ the thickness, P the internal pressure, EK Young’s 
1 : : : k > 
Modulus, a Poisson’s Ratio, and z the radial displacement at any point x measured 
along the axis from some fixed origin, the longitudinal stress being assumed zero. 
The solution of the equation is 
2 
z= cos nx (A cosh nz + B sinh nx) + sin nx (C cosh na + D sinh nx) + = Sal), 
4 Or (een +985 
Se Peroni? 1: = if : for steel be taken as 0-3. If 7 be the 
where 2 = ass syste 
m* 4/tR m 
SS 
LEDILEDEIET IE LE ELISSLITOLLOEL 1S OE EE EIEIO ILE 
I Eo OL aa ca ao 
Ss Ss RSs SS 
Fie. 1 
length of the pipe between successive rings (fig. 1) and 2 be measured from A as 
origin, then, assuming that the pipe wall remains cylindrical under each ring, 
© =0 fore =0 andz2=1 
and 2= 2, tor 2—0 and e¢—7, 
where 2, is the radial displacement at the ring. 
Evaluating the constants A, B, C, D, equation (1) becomes 
2 D 2 
be — — ea — Ze) (cosh nx cos naz — H sinh na cos nz + H cosh nx sin nx 
—Lsinhnesinnz) . é : erinu (2) 
j cosh nl—cosnl , — sinh nl — sin nl 
wee ~ sinh nl + sin nl’ ~~ sinh nf + sin nl 
In determining the radial displacement at the ring it will simplify the work if, 
as will always be the case in practice, the radial thickness h of the ring is small com- 
pared with R, and the small error involved in taking R to be the radius of the 
ring be neglected. 
Considering the portion of the pipe in contact with the ring, the forces acting 
upon it in the radial direction will be (1) the internal pressure, (2) the external 
pressure produced by the ring, and (3) the shearing force across the pipe wall at each 
side of the ring. Denoting by P the pressure per unit area exerted by the ring on 
