sm8 31 33

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PROBLEMS (8.31 through 8.33) At three points in a structure subjected to plane stress, the
stresses have the magnitudes and directions shown in Figs. P8.31 through P8.33. For each point, obtain
the principal stresses and the related orientations. Show the results on properly oriented elements.
SOLUTION (8.31)
MPaMPaMPa xyyx 4,2,3
=
==
τ
σ
σ
22
1,2 32 32
( ) 4 2.5 4.03
22
σ
+−
=± +=±
MPaMPa 53.153.6 21
−
==
σ
σ
12(4)
2 tan 82.9 ; 41.4
32 oo
pp
θθ
−
== =
−
'3232
cos82.9 4sin82.9
22 oo
x
σ
+−
=+ +
1.53 MPa
41.4o
6.53 MPa
x
MPa53.6=
Thus,
' 41.4o
p
θ =
SOLUTION (8.32)
ksiksiksi xyyx 1,6,15
=
==
τ
σ
σ
22
1,2 15 6 15 6
( ) 1 10.5 4.61
22
σ
+−
=± +=±
ksiksi 89.511.15 21 ==
σ
σ
12(1)
2 tan 12.5
15 6 o
p
θ
−
==
−
'15 6 15 6 cos12.5 1sin12.5
22 oo
x
σ
+−
=+ +
5.98 ksi
6.3o
15.11 ksi
x
ksi11.15=
Thus,
'6.3
o
p
θ =
Continued on next slide
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SOLUTION (8.33)
MPaMPaMPa xyyx 2,6,14
−
=
==
τ
σ
σ
22
1,2 14 6 14 6
()(2)104
22
σ
+−
=± +−=±
.5
MPaMPa 5.55.14 21 ==
σ
σ
12( 2)
2 tan 26.6
14 6 o
p
θ
−−
==−
−
'14 6 14 6 cos( 26.6 ) 2sin( 26.6 )
22 oo
x
σ
+−
=+ −+−
=
MPa5.14
Thus,
' 13.3o
p
θ =−
5.5 MPa
13.3o
14.5 MPa
x
Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or
instructional purposes only to students enrolled in courses for which the textbook has been adopted. Any other
reproduction or translation of this work beyond that permitted by Sections 107 or 108 of the 1976 United States
Copyright Act without the permission of the copyright owner is unlawful.