sm8 43

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PROBLEM (*8.43) A structural member is subjected to two different loadings, each separately
producing stresses at point A, as indicated in Fig. P8.43. Calculate, and show on a sketch, the principal
planes and the principal stresses under the effect of the combined loading.
*SOLUTION
10 ksi
θ
=-30o
20 ksi
x
A
10 20 10 20 cos( 60 ) 15 2.5 12.5
22 o
xksi
σ
+−
=+ −=−=
ksi
y5.175.215 =+=
σ
10 20sin( 60 ) 4.33
2o
xy ksi
τ
−
=− − =−
17.5 ksi
12.5 ksi
4.33 ksi
A
A
5 ksi
17.5 ksi
12.5 ksi
0.67 ksi
+
=
A
22
1,2 12.5 17.5 12.5 17.5
( ) 0.67
22
σ
+−
=± + 59.215
±
=
12
17.59 12.41ksi ksi
σ
σ ==
12(0.67)
2tan[ ]1
12.5 17.5 o
p
θ
−
==
−5−
'12.5 17.5
15 cos( 15 ) 0.67sin( 15 )
2oo
x
σ
−
=+ − + −
=− 15 2.41 0.173 12.41 ksi− =
Thus,
"7.5
o
p
θ =−
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12.41 ksi
17.59 ksi
7.5o
x
A