36
2 Analysis of Stress
∴ The cubic equation becomes
σ
3
− 18σ
2
+ 52σ − 27 = 0
The roots of the cubic equation are the principal stresses. Hence, the three principal
stresses are
σ 1 = 14.554 MPa; σ 2 = 2.776 MPa and σ 3 = 0.669 MPa
Now to find principal directions for σ 1 stress:
(9 − 14.554)
6
3
6
(5 − 14.554)
2
3
2
(4 − 14.554)
=
−5.554
6
3
6
−9.554
2
3
2
−10.554
A =
−9.554
2
2
−10.554
= 100.83 − 4 = 96.83
B = −
6
2
3 −10.554
= −(−63.324 − 6) = 69.324
C =
6 −9.554
3
2
= 12 + 28.662 = 40.662
A 2 + B 2 + C 2
=
(96.83)
2
+ (69.324)
2
+ (40.662)
2
= 125.83
l 1 =
A
√
A 2 + B 2 + C 2
=
96.53
125.83
= 0.769
m 1 =
B
√
A 2 + B 2 + C 2
=
69.324
125.83
= 0.550
n 1 =
C
√
A 2 + B 2 + C 2
=
40.662
125.84
= 0.325
Similarly, the principal stress directions for σ 2 stress and σ 3 stress are calculated.
Therefore,
l 2 = −0.226 l 3 = 0.596
m 2 = −0.177 m 3 = −0.800
n 2 = 0.944 n 3 = 0.057