20 The N.Z. Journal of Science and Technolog-y. [Feb. 
plates of equal thickness. (Requirement (1) can be attained if the bottom 
plate or plates are made of thinner material than the others, a possibility 
which has not been recognized hitherto.) The natural device suggests 
itself of trimming the overhanging ends of each plate, either in width or 
thickness, or both, and varying the overhangs until these requirements 
are approximately arrived at, and this summarizes the latest development 
in spring-design as explained by Landau and Parr. Unfortunately, how¬ 
ever, the fact has been entirely overlooked that these formulae neglect the 
deflection caused by shear, and that, while this deflection is small absolutely, 
it has a considerable effect upon the values of the successive loads and 
stresses in the plates. 
The deflection due to shear in a uniform cantilever is incorrectly treated 
in some of the well-known text-books, and the correct calculation is re¬ 
peated here in full. Since the shearing-force is uniform, the researches of 
St. Venant* * * § show that Bernouilli’s assumptions hold, and therefore the 
shearing-stress across any plane normal to the neutral axis of a rectangular- 
section cantilever may be expressed! as 
6W ft 2 
q= W\T~ y 
where y is the distance of the element above or below the neutral axis. 
The internal work done in producing the shearing-strain, since the 
q 2 
material is elastic, is ~ at any place (where C is the modulus of shear 
elasticity). Therefore the internal shear-work done along the cantilever is 
r r , 36Z6W 2 ri [t± thy 2 
2C 
2CbH 6 
1SIW 2 ft 5 
16 
C W 
16 24 ^ 80/ 
But this must equal the external work 
w y 
due to shear. Therefore 
2 + y i j dy 
3W n 
5C bt 
where y is the deflection 
y» = 
6WZ 
5C bt 
4 bt^ 
For steel of spring quality, C may be taken as -- E, and since I = ——. 
10 1 2 
y s 
wit 2 
4EI 
A further correction still remains to be made. When a beam is bent 
downwards the upper layers are in tension and the lower layers in com¬ 
pression ; therefore the upper layers contract laterally and the lower layers 
expand. These lateral strains are — times the longitudinal strains, where — 
m m 
is Poisson’s ratio, and if unconstrained the beam is curved laterally to a 
radius m times the radius of longitudinal curvature. Narrow beams do 
so curve, but in broad flat strips no transverse curvature occurs, and the 
modulus of elasticity is altered in the proportion 
m- 
m 2 — 1' 
That means that 
flat strips are stiffer than the theoretic beam in the above proportion. Now, 
* Arthur Morley, Strength of Materials (3rd ed.), p. 107. 
f Arthur Morley, loc. cit., p. 134. 
$ J. A. Ewing, The Strength of Materials (2nd ed.), p. 144. 
§ Arthur Morley, loc. cit., p. 233. 
