Mohr's Circle: 3D Stress Analysis Solution

_______________________________________________________________________
PROBLEM (8.76) An element in three-dimensional stress is subjected to stresses
x
σ
= 120 MPa
y
σ
= 60 MPa,
x
y
τ
= -30 MPa, z
σ
= -15 MPa
as shown in Fig. P8.76. Using Mohr’s circle, determine:
(a) The principal stresses.
(b) The absolute maximum shear stress.
SOLUTION
1()
2
avg x y
σ
σσ =+
1(120 60) 90
2
M
Pa=+=
22
()
2
xy
x
y
R
σσ
τ
−
=+
22
120 60
()(3
2
−0)−=+
42.43
M
Pa
=
O
σ2
(a) 1,2 avg
R
σ
σ =±
190 42.43 132.4
M
Pa
σ
=+ =
247.6
M
Pa
σ
=
315
M
Pa
σ
=−
(b) max 1 3
1
() (
2
a)
τ
σσ =−
1[132.4 ( 15)] 73.7
2
M
Pa=−−=
C
σ3
σavg =42.43
τ (MPa)
σ
(MPa)
σ
1
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