3D Stress Problem Solution: Max Shear Stress Calculation

_______________________________________________________________________
PROBLEM (8.80) A point in a machine component is subjected to three-dimensional stress with
x
σ
= 150 MPa y
σ
= 30 MPa
x
y
τ
= 90 MPa z
σ
as depicted in Fig. P8.105. Determine:
(a) The maximum shear stress for z
σ
= 40 MPa.
(b) The maximum shear stress for z
σ
= - 40 MPa.
SOLUTION
1()
2
avg x y
σ
σσ =+
1(150 30) 90
2
M
Pa=+=
22
()
2
xy
x
y
R
σσ
τ
−
=+
22
150 30
()9
2
−0=+
108.2
M
Pa
=
1198.2
avg
R
MPa
σ
σ
=
+=
318.2
M
Pa
σ
=
−
(a) 21 3 40 198.2 18.2
M
Pa MPa MPa
σ
σσ == =−
max 1 3
1
() (
2
a)
τ
σσ =−
1[198.2 ( 18.2)] 108.2
2
M
Pa=−−=
(b) 31 2
40 198.2 18.2
M
Pa MPa MPa
σ
σσ =− = =−
max 1 3
1
() (
2
a)
τ
σσ =−
1[198.2 ( 40)] 119.1
2
M
Pa=−−=
R
σavg =90
τ (MPa)
C
σ
(MPa)
σ
σ1
3
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