sm9 98

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PROBLEM (9.98) A steel shaft of yield strength y
σ
and diameter d is subjected to an axial load
P, torque T, and bending moment M as portrayed in Fig. P9.98. Calculate the safety factor
s
n with
respect to yield based on the maximum energy of distortion criterion of failure.
Given: d = 40 mm, P = 20 N, T = 500 N
⋅
m, M = 350 N
⋅
m, y
σ
= 250 MPa
SOLUTION
The critical stress occurs at a point A.
xPMc
AI
σ =+
23
432PM
dd
ππ
=+
23
4(20) 32(350) 71.62
(0.04) (0.04)
M
Pa
ππ
=+=
33
16 16(500) 39.79
(0.04)
xy T
M
Pa
d
τππ == =
Principal stresses are then
22 2 2
1,2 ( ) 35.81 (35.81) (39.79)
22
xx
xy
σσ
στ
=± += ± +
35.81 53.53=±
12
89.34 17.72
M
Pa MPa
σ
σ ==−
Therefore
222
1122
();
y
s
n
σ
σσσσ −+=
2
22
2
(250)
(89.34) (89.34)( 17.72) ( 17.72)
s
n
−−+−=
or
2.51
s
n=
x
y
τ
A
y
x
x
σ
T
M
x
A
P
d