Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

SolutioN
The ultimate strength σ U for heat-treated SAE 4340 steel (see Appendix D for its properties)
is 1,030 MPa. Thus, the allowable stress for the spring is
σ U
1, 034 MPa
σ allow = = = 413.6 MPa
FS 2.5
The moment of inertia at the minimum spring depth is
3
(50mm)(40mm)
4
I = = 266,667 mm
12
An expression for the allowable bending-moment magnitude in terms of the stressconcentration
factor K can be derived from Equation (8.26) by setting σ max = σ allow and
σ nom = M allow c/I:
σ max σ allow
K = =
σ M c
nom allow
I
Solve this equation for M allow to derive
M
allow
2 4
σ allowI
(413.6 N/mm )(266,667 mm ) 5,514,574 Nmm ⋅ 5,515 Nm ⋅
= = =
=
Kc K(20mm)
K K
With reference to the nomenclature used in Figure 8.18, the ratio of the maximum spring
depth D to the reduced depth d is D/d = 80/40 = 2.0.
(a) Fillet Radius r = 4 mm
A stress-concentration factor K = 1.84 is obtained from Figure 8.18 with D/d = 2.0 and
r/d = 4/40 = 0.10. The maximum allowable bending moment is thus
5,515 Nm ⋅ 5,515 Nm ⋅
M =
=
= 2,997 Nm ⋅ Ans.
K 1.84
(b) Fillet Radius r = 12 mm
For a 12 mm fillet, r/d = 12/40 = 0.30, and thus, the corresponding stress-concentration factor
from Figure 8.18 is K = 1.38. Accordingly, the maximum allowable bending moment is
M
=
5,515 Nm ⋅ 5,515 Nm ⋅
=
K 1.38
= 3,996 Nm ⋅ Ans.
pRoBLEmS
p8.69 A stainless steel spring (shown in Figure P8.69/70) has a
thickness of 3/4 in. and a change in depth at section B from D = 1.50 in.
to d = 1.25 in. The radius of the fillet between the two sections is r =
0.125 in. If the bending moment applied to the spring is M =
2,000 lb ⋅ in., determine the maximum normal stress in the spring.
p8.70 A spring made of a steel alloy (shown in Figure P8.69/70)
has a thickness of 25 mm and a change in depth at section B from
D = 75 mm to d = 50 mm. If the radius of the fillet between the two
sections is r = 8 mm, determine the maximum moment that the
spring can resist if the maximum bending stress in the spring must
not exceed 120 MPa.
(1)
D
A
FIGURE p8.69/70
B
d
Radius r
(2)
C
M
305