sm9 66

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_______________________________________________________________________
PROBLEM (*9.66) Loads P and R are applied at the free end of the post having a diameter d shown
in Fig. P9.66. Calculate the principal stresses and the maximum shearing stress:
(a) At point A.
(b) At point B.
Indicate the results on properly oriented elements.
Given: P = 400π⋅N, R = 800π⋅N, d = 50 mm
*SOLUTION
mmd 50
=
NP
π
400
=
NR
π
800
=
(a) 33
16 16(0.4 400 )
(0.05)
tT
d
π
τππ
×
== MPa48.20
=
23
800 32(0.6 400 )
(0.025) (0.05)
z
xMy
R
AI
π
π
σππ
×
=+ =− −
MPa7.6244.6128.1 −=−−=
22
1,2 62.7 62.7
( ) (20.48)
22
σ =− ± − +
45.3735.31 ±−=
MPaMPa 8.681.6 21 −==
σ
σ
MPa5.37
max =
τ
2 2(20.48)
tan2 ' , ' 16.6
62.7 o
pp
x
τ
θθ
σ
== =−
−
Continued on next slide
A
x
x
σ
t
τ
T=0.4P
y
z
R
M
z=0.6P
M
y=0.4R
x
P
A
C
B

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(b)
2
44
33
dVQ V P
I
bAr
τ
π
== = (Example 9.4)
2
4 400 0.85
3 (0.025)
M
Pa
π
π
==
and
MPa
dt 33.2185.048.20
=
+=+=
τ
τ
τ
Similarly,
3
32(0.4 800 )
1.28 (0.05)
x
π
σπ
×
=− −
MPa2.8392.8128.1 −=−−=
Continued on next slide
B
x
y
x
M
z
R
AI
σ =+
td
τ
ττ
=
+
37.5 MPa
6.1 MPa
16.6o
68.8 MPa
x
y
τ (MPa)
σ
(MPa)
C
'
2
p
θ
(−62.7, −20.48 ) 1
σ
2
σ
A

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Thus,
22
1,2 83.2 83.2
( ) (21.33)
22
σ =− − +m
75.466.41 m−=
MPaMPa 4.8815.5 21
−
==
σ
σ
MPa8.46
max =
τ
2 2(21.33)
tan2 ' 83.2
px
τ
θσ ==
−, ' 13.6o
p
θ =−
46.8 MPa
5.15 MPa
13.6o
48.4 MPa
B