sm9 102

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PROBLEM (9.102) At a particular point in a cast aluminum alloy ( u
σ
= 10 ksi, '
u
σ
= 26 ksi)
structure subjected to plane stress, the stresses have the magnitudes and directions shown in Fig. P9.102.
Determine whether failure occurs at the point according to:
(a) The maximum principal stress theory.
(b) Coulomb-Mohr theory.
SOLUTION
We have 25 30 0.
yxyx
ksi ksi
σ
τσ =− = =
Thus
22 2 2
1,2 () 3 (3)(8)
22
yy
xy
σσ
στ
=− ± + =− ± +
12
5.54 11.54ksi ksi
σ
σ ==−
(a) Maximum principal stress theory:
11; 5.54 10
u
σ
σ
=<
But
21; 11.54 10
u
σ
σ
=> (failure occurs)
(b) Coulomb-Mohr Theory:
12
'5.54 11.54
1; 1
10 26
uu
σ
σ
σσ
−
−= − =
gives
0.554 0.443 0.998 1+=< (no fracture)
Comment: Coulomb-Mohr criteria is more reliable when '
uu
σ
σ >> , as in this
problem.