1920 .] 
Jenkinson.—Design of Laminated Springs. 
23 
indicated in fig. 5. The deflection due to bending in this tapered canti¬ 
lever is calculated as follows 
/ a 
d 2 y — Wx_ — W xh 
dx 2 EIz Elba; 
W xb 
War 
f 
r — 1 
EI \z + (b — EI(f 
- 1 
r/ a 
r — 1 
Integrating 
War 
dx 
but 
Therefore 
dy War 
dy 
dx EI (r — 1) i r — 1 
dy 
log (““I + ») — ® + Kf 
r — 1 
0 when x = a., so K = a 
a 
dx EI (r - 1) ( r 
i log ( 7 h + 
x — 
a . ar 
l0 « rW 1 
a . ar 
log- - — x + a 
r — 1 b r — 1 
and 
y = 
War 
a 
EI (r- 1) 1 r 
— ( - °—r + X ) log ( - °—r + 
1 \r — 1 1 / 6 Vr — 1 
x 
a 
r— 1 
x 
a ar ar I ) 
- JZTi °8 7^7 * - - + ax + Kj 
/ a \ 2 a 
But y — 0 when x = 0, so K = — i———J log —- - 
Therefore 
y 
War 
a 
EI (r — 1) |_\r—1/ 1 °\r—1 
a f , (a 
+ 7 = 1*1 lo 8 (— 1 + 
and finally integrating between limits 0 and a 
x 
lo § (77+*7 lo g ~i | 
2 
1 — lo S “ - 2 
] 
a 
a- 
a■ 
db = bit - 1) 1771 k> 8 r -7=i+T-j 
Wa 3 f r , r (r — 3) ) 
= “ET I (r — l) 3 log r + 2 (r— l) 2 f 
and introducing the correction for lateral stiffness 
dh 
15W a 3 
r . , r (r — 3 
- 7 log r + 
16EI 1 (r — l) 3 ' ‘2 (r — l) 2 ) 
The deflection due to shear is calculated as follows :— 
6W ft 2 
As before, the intensity of shear stress q x — (-j — y 2 ), and internal 
work done in producing the shear strain is 
t.e., 
2C Jo 
3W 2 [ a dx 
q x 2 dx dy 
3W 2 ar 
5C t Jo b 7 _ b\ x 5C tb {r - 1) 
log r 
r a 
