sm8 45

_______________________________________________________________________
PROBLEM (*8.45) Redo Prob. 8.43 for the case shown in Fig. P8.45, where
σ
and
θ
are
known constants.
*SOLUTION
Note:
cos()cos
θ
θ
−
=
sin()si
σ
A
n
θ
θ
−
=−
θ
−
x
2
cos2( ) (1 cos2 ) cos
22 2
x
σ
σσ
σ
θθσ =+ −= + = θ
2
cos2( ) (1 cos2 ) sin
22 2
y
σ
σσ
σ
θθσ =− −= − = θ
sin2( ) sin cos
2
xy
σ
τ
θσθ =− − = θ
A
σ
+
=
2
sin
σ
θ
sin cos
σ
θθ
2
cos
σθ
A
2
(1 sin )
σ
θ +
sin cos
σ
θθ
2
cos
σ
θ
A
For the combined stress:
22
'(cos1sin
22
xy )
σ
σσ
σ
θθ
+
== ++
σ =
22
(cos 1 sin ) sin
22
xy 2
σ
σσ
θ
θσ
−=−−=−
θ
2
sin cos
tan2 sin
p
σ
θθ
θσθ
==−
−cotan
θ
290; ' 45
2
oo
pp
θ
θθθ =+ =+
2222222
)() sin(sincos
2
xy xy
r
σ
σ
τ
σθθ θ
−
=+= +
22
sin
σ
θ =
Thus,
)sin1(sin'
1
θ
σ
θ
σ
σ
σ
σ
+
=
+=+= r
)sin1(sin'
2
θ
σ
θ
σ
σ
σ
σ
−
=
−=−= r
A'
p
θ
1
σ
2
σ
x
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