24 
The N.Z. Journal of Science and Technology. 
[Feb. 
And since this must equal the external work 
6W ar Wat 2 r , 
s = 5C tb (r — 1) ° g r = 4EI (r— 1) ° g ’ 
so that total deflection is given by 
15Wa 3 r ( log r r — 3 4£ 2 
16EI (r — 1) ((r — 1) 2 2 (r — 1) 15a 2 ? 
But the deflection in the same length of leaf immediately above is 
given by 
7 15Wa 3 
f ~ 32EI 
and the equation for r is 
r ( log r ,4£ 2 r — 3 I 
-■ --- -4-log f -4- -?- 
r — 1 (. (r — l) 2 ' 15a 2 8 ^2(r-l) ( 
and a real positive value of r can be found by trial, remembering that the 
logarithms are Napierian ones. 
Springs can therefore be designed on this basis. The first step is to 
determine a by means of the formula 
Wa = 
S bt 2 
~(T 
, where W = J total load in pounds. 
Then by equation (8) we can determine r by trial, since t and a are known. 
In order to have a factor of safety should any of the upper plates break, 
it is usual to make about one-fourth of the whole number of leaves full 
length, and full width for their whole length. 
Assuming that m full-length leaves are used and n shorter ones, the 
length of the cantilever (that is, half the span of the spring—half the 
width of the buckle) is (m -f- n) a, and each of the overhangs equals a, 
except that for the m top plates of full length, which is ma. 
We can now calculate the deflection of the spring by equation (6), 
since 
or 
15W ((m-f-w) 3 a 3 (m + n)n 2 a 3 n 3 a 3 ) Wat 2 
16EIm (3 2 6 1 4EI 
15Wa 3 (3 (m -J- n ) 2 — m 2 4 1 2 i 
16ET ( 6 ^ 15a 1 1 
( 9 ) 
The number of leaves, and consequently the span of the spring, is chosen 
to give such deflection as is desirable. 
To make this portion of the paper complete, it may be stated that E can 
be taken as 30,000,000 for spring steel, that the maximum allowable fibre 
stress S usually varies from 60,000 lb. to 80,000 lb. with ordinary carbon 
spring steel, and that in locomotive practice the deflection under working¬ 
load is usually about 2 ±n. 
Fig. 6 shows a spring designed in this manner. The approach of such a 
spring to theoretic perfection is very close. Each of the plates is stressed 
to the same amount, and the plates are stressed to this maximum through¬ 
out nearly the whole of their length. The only sensible variation from this 
requirement is due to the fact that the m top plates are not tapered for the 
amount of the overhang, but are carried out full width. As equation (9) 
shows, this has only a very small effect on the deflection, unless m is greater 
than one-half the whole number of plates. The amount of spring steel 
