sm8 29

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PROBLEM (8.29) An element in plane stress is subjected to stresses with
x
σ
= 65 MPa, y
σ
= 30
MPa, and
x
y
τ
=25 MPa (Fig. P8.29). Calculate:
(a) The principal stresses.
(b) The maximum shear stresses and associated normal stresses.
Sketch results in a properly oriented element.
SOLUTION
(a) 22(25)
tan2 65 30
xy
pxy
τ
θσσ ==
−−
or
and
25
o
p
θ = 5 27.5o
p
θ =
11
' ( ) (65 30) 47.5
22
avg x y
M
Pa σσ σσ == += +=
1
2
22
[( ) ]
2
xy xy
R
σ
στ −
=+
1
2
22
65 30
[( ) 25 ] 30.5
2
M
Pa
−
=+ =
1,2 47.5 30.5
avg
R
MPa
σ
σ =±= ±
Thus,
12
78 17
M
Pa MPa
σ
σ ==
Note, for
27.5
p
θ =:
o
'11
()()cos2sin2
22
x
xy xy pxy p
σ
σσ σσ θτ θ =++− + , gives
'78
x
M
Pa
σ
=
(b) max 30.5 ' 47.5
R
MPa MPa
τ
σ == =
x
’
30.52 MPa 27.5o
78 MPa
x
17 MPa
47.5 MPa
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