22 The N.Z. Journal of Science and Technology. [Feb. 
But if the stresses in these plates are equal, by (7) 
W 2 = W,|l 
and 
_ 1(V - 16 l x t 2 
2 _ ioi, 2 - m * 
This gives us a spring in which all the leaves are the same length, and 
the reason for this deadlock is the fact that the bottom leaf is always 
stressed more than the others in any spring with square-ended leaves 
of equal section. The only way to relieve this overstress is to increase 
the flexibility (reduce the stiffness) of the bottom leaf, and the simplest 
way to do this is to reduce the thickness. Fig. 1 shows a spring de¬ 
signed in this manner, where the majority 
of the leaves are in. thick and the 
bottom leaf is composed of thinner plates 
of correct length. The stresses in this 
spring are equal in every plate at the 
buckle, but decrease to zero at the 
point, so that, while the spring is of 
maximum strength for the number of 
leaves used, it is not by any means so 
flexible as it might be. To obtain a 
spring of maximum flexibility the stress 
must be maintained at its maximum 
allowable value throughout the whole 
length of the leaf, and this cannot 
possibly be arrived at with square-ended 
leaves. 
The next step obviously is to test whether leaves with taper ends 
can be designed to give the desired results. If we assume that plates 
of the same section are used, and that the stresses are equal in each plate 
at the buckle and at the point where the plate beneath supports it, then 
by equation (7) 
W n -f 1 4 + 1 W 'fjjn — W n l n _ i l n — i = . &C. 
= -j -1 (l n + i l n ) ==z W n ( l n — i) — . &C. 
trom which we deduce 
W n + ! = W n = W n _ 1 . &c., and 
In + 1 === hi In — 1 =ir ~ . &C., 
or the overhangs and reactions are equal throughout, and the bending- 
moments are also equal in the non-tapered part of each leaf, while the 
tapering extends for the whole length of the overhang and no more. 
Further, there is no shear stress in any part of the leaf save the overhang, 
so that the greater portion of the spring is bent to a constant radius of 
curvature. 
At first sight it appears that we are back at the “ theoretic ” spring of 
fiction, but it is not so. The problem now is to find whether the “ overhang ” 
can be so tapered that the deflection therein due to bending and shear is 
equal to the deflection in the same length of the leaf above it. As tapering 
in width is cheaper than tapering the thickness, let us assume that the leaf 
is tapered from full width h to width - in the length a of overhang, as 
X 
Fig. 5. 
