sm8 80

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PROBLEM (8.80) A point in a machine component is subjected to three-dimensional stress with
σx
= 150 MPa
σy
= 30 MPa
τ xy
= 90 MPa
σz
as depicted in Fig. P8.105. Determine:
(a) The maximum shear stress for σ z = 40 MPa.
σz
(b) The maximum shear stress for
= - 40 MPa.
SOLUTION
1
2
1
= (150 + 30) = 90 MPa
2
σ −σ y 2 2
R= ( x
) + τ xy
2
σ avg = (σ x + σ y )
τ (MPa)
σavg =90
σ3
σ1
C
150 − 30 2
) + 90 2
= (
2
= 108.2 MPa
σ (MPa)
R
σ 1 = σ avg + R = 198.2 MPa
σ 3 = −18.2 MPa
(a) σ 2 = 40 MPa
σ 1 = 198.2 MPa
σ 3 = −18.2 MPa
1
(τ max ) a = (σ 1 − σ 3 )
2
1
= [198.2 − (−18.2)] = 108.2 MPa
2
σ 1 = 198.2 MPa
σ 2 = −18.2 MPa
(b) σ 3 = −40 MPa
1
(τ max ) a = (σ 1 − σ 3 )
2
1
= [198.2 − (−40)] = 119.1 MPa
2
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