1920 .] 
Jenkinson.—Design of Laminated Springs. 
21 
for hard steel the modulus of shear elasticity has been taken as — E, from 
which it follows that m is 4, and broad flat beams are stifler than the 
theoretic beam in the proportion of 16 to 15. We can now modify equa¬ 
tions (1), (2), and (3) to include both these refinements, and write the true 
deflections as 
d = - y n + y s and 
d n — 
15 
W„ 
16EI 
dn + 1 = d n + 
n + 1 \ l n + 1 
l 
15 (l 
2 6 
'n + 1 In) 
16 
-1-VWn 
7 2 7 3 \ 7 3 
fcw. h n \ ~wj hi 
+ 
(W : 
n + 1 
W ft ) 
4EI 
..( 4 ) 
16E1 
W. 
n + 1 \ v n + 
+ 1 4 
In 2 , fin + l In) 2 
+ 
/ 2 
l n 
-W n’^r + 
4W 
n + 1 
t 2 
15EI ) 
( 5 ) 
■'n + 1 — 
'n + 
15 ( w l n +1 3 
16EI i W “ +1 3 W “ V 2 
, (W» + l l n +l — W rJ'n)t 2 
] 2 
1 f'n 
1 3 
b n 
4EI 
( 6 ) 
' These equations are true only for square-ended plates, and are subject 
to suitable modifications where plates of varying width or thickness are 
used or when more than one plate is made the same length. 
These equations can be used to evaluate the various reactions in the case 
of existing springs, with the consequent stresses in the various plates. When 
new springs are being designed the stresses are made equal at the buckle 
by suitably proportioning the length (and thickness) of the various leaves, 
using the further equation 
n + 1 n + 1 
W 7 W 7 _ W 7 
V¥ n b n yy n L n yy n - 1 b n - 1 
U‘ 
bt■ 
id) 
In practice the section of spring steel used in making laminated springs 
is not exactly rectangular, but has rounded edges. This fact may increase 
the deflection by a small amount—in the neighbourhood of 5 per cent.— 
but has practically no effect on the relative value of the deflections expressed 
by equations (4), (5), and (6). If deemed necessary it can be taken care of 
by a proportionate reduction in the value of I, but this is a refinement 
hardly warranted by the accuracy of the experimental knowledge. The 
important point is not the absolute value of the deflection, but the accuracv 
of the relative values expressed by the equations and used for calculating 
the successive reactions and stresses. 
We are now in a position to consider the design of a simple spring with 
square-ended plates. 
Applying equation (6) to the bottom leaf and (4) to the second leaf, 
we have 
15 WA 3 , WA« 2 
= 16 "T" + 4 
BW i = 7c 
16 
16 
W 
(hh 2 h 3 \ 
2 \ 2 6 7 
L 3 ' 
3 
W, v] + 
(w.-wju 
whence 
W 2 = W 1 
2oi, 2 + m' 
15 l 2 l 1 — 5^ 2 + 8 1' 
