sm8 15

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PROBLEM (8.15) At a point in a loaded structure, the normal and shear stresses have the magnitudes
and directions acting on the inclined element depicted in Fig. P8.15. Calculate the stresses
x
σ
, y
σ
, and
x
y
τ
on an element whose sides are parallel to the xy axes.
SOLUTION
Transform from 3 to 0o
θ =0
θ
=
. For convenience in computations, Let
200 , 60 , 35
xyxy
M
Pa MPa MPa
σ
στ =− =− = and 30o θ =−
Then
'11
()()cos2sin
22
xxyxy xy
2
σ
σσ σσ θτ θ =++− +
11
( 200 60) ( 200 60)cos( 60 ) 35sin( 60 )
22 oo
=−−+−+ − + −
195.3
M
Pa=−
'' 1()sin2cos
2
xy x y xy 2
τ
σσ θτ θ =− − +
1( 200 60)sin( 60 ) 35cos( 60 )
2oo
=− − + − + −
43.1
M
Pa=−
So
''
200 60 195.3 64.7
yxyx
M
Pa
σ
σσσ =+− =− −+ =−
For 0
θ =:
o
6 MPa4.7
195.3 MPa
3.1
4 MPa
y
x
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