1920.] 
Jenkinson.—Design of Laminated Springs. 
17 
In 1909 Professor Unwin in the new edition* of his famous.work used the 
fundamental assumption that the leaves are in contact only at the ends, 
but failed to make any real use of the assumption. Finally, in 1918, David 
Laudau and Percy H. Parrf published an elaborate and laborious investiga¬ 
tion of the whole subject, in which the theory underlying the anonymous 
article in The Engineer of 1902 was exhaustively treated, but without 
noticing the incompleteness that vitiated all their conclusions. 
If a cantilever of length l, width h, and thickness or depth t, is loaded with 
a weight W at its end, we know that 
M _ E _ S _ 2s 
I R y t 
so that the radius of curvature of bending at any point is given by 
Now, the efficiency of the cantilever as a spring is- measured by its- 
deflection, and this is greatest when R is least. Taking E, the modulus of 
elasticity, as a constant for the material, it is plain that the spring is most 
efficient when t is as small as practicable and when S, the unit fibre stress, 
is the maximum allowable throughout the whole length of the cantilever. 
Since % _ 6M 
“ T ~ 6C 
it is clear that bt 2 should vary as M, the bending-moment, throughout the 
cantilever. It is simpler and cheaper to shape a plate to a variable width 
than to a variable thickness (especially where, as in this case, the curve of 
thickness is parabolic in shape), so we arrive at a flat triangular plate as the 
shape for a spring. In practice, however, space considerations prevent the 
use of this form, and the single flat plate is replaced by a number of 
narrower strips which are piled one below the other and held together by 
the central buckle. If the mathematical design outlined above is followed 
the triangular flat plate reappears in the successive leaves, and each over¬ 
hang will be equal in length and triangular in plan, as indicated in fig. 2. 
Such a spring is the spring that is referred to in the text-books as the 
“ theoretic ” spring, but a very little consideration will show that this misuse 
* W. C. Unwin, Elements of Machine Design (1909), pt. 1, p. 104. 
t David Landau amd Percy H. Parr, A New Theory of Plate Springs, Journal 
of the Franklin Institute , vol. 185, pp. 481-507 ; vol. 186, pp. 699-721 ; vol. 187, 
pp. 65-97 and pp. 199-213 (April and December, 1918; January and February, 1919). 
2—Science. 
