_______________________________________________________________________
PROBLEM (9.22) A steel helical spring fits within a brass helical spring (see Fig. 9.7). Each spring
has ends constrained to deflect an identical amount. Calculate:
(a) The total permissible load P the two springs can sustain jointly.
(b) The ratio of the spring rates.
Given: The properties of each spring are as follows.
----------------------------------------------------------------------------------------------------------
Outer (brass) spring Inner (steel) spring
------------------------ -----------------------
Mean diameter, 2R 200 mm 140 mm
Wire diameter, d 20 mm 20 mm
Number of coils, N 10 10
Modulus of rigidity, G 40 GPa 80 GPa
Allowable shear stress, all
τ
150 MPa 250 MPa
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SOLUTION
(a) sb
δ
δ
=
333
44
64 64 ;
bb ss bb ss
bsb
NP R NPR P R PR
dG dG G G
==
3
s
Substitute the given data:
33
99
(0.1) (0.07) ,5.8
40 10 80 10
bs 3
s
b
PP PP
==
××
Assume steel controls:
63
16 (0.07) 20
250 10 (1 )
(0.02) 4 70
b
sP
τπ
=×= +
×
5.24 0.9
sb
PkNP==kN
Check steel spring's stress:
3
3
16(0.9 10 )(0.1) 20
(1 )
(0.02) 4 100
b
τπ
×
=+
×
60.2 150
M
Pa MPa=< O.K.
Thus,
Pk
5.24 0.9 6.14
total N=+=
(b) 44
33
,
64 64
bbb s
bs
bbb
PPdG PdG
kk
NP R NPR
δ
== = s
ss
Therefore,
33
380 100
( ) 5.83
40 70
ssb
bbs
kGR
kGR
== =
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