50
2 Analysis of Stress
= 200 + 270.3
C = 470.3
A 2 + B 2 + C 2 =
(645.29)
2
+ (646) 2 + (470.3) 2
= 1027.08
∴ l 1 =
A
√
A 2 + B 2 + C 2
=
645.29
1027.08
= 0.628
m 1 =
B
√
A 2 + B 2 + C 2
=
646
1027.08
= 0.628
n 1 =
C
√
A 2 + B 2 + C 2
=
470.3
1027.08
= 0.458
(b) To find principal plane for stress σ 2
(10 + 10)
20
10
20
(10 + 10)
10
10
10
(10 + 10)
=
20 20 10
20 20 10
10 10 20
A =
20 10
10 20
= 400 − 100 = 300
B = −
20 10
10 20
= −(400 − 100) = −300
C =
20 20
10 10
= (200 − 200) = 0
A 2 + B 2 + C 2 =
(300)
2
+ (−300) 2 + (0) 2 = 424.26
∴ l 2 =
A
√
A 2 + B 2 + C 2
=
300
424.26
= 0.707
m 2 =
B
√
A 2 + B 2 + C 2
=
−300
424.26
= −0.707
n 2 =
C
√
A 2 + B 2 + C 2
= 0
(c) To find principal plane for stress σ 3
(10 − 2.7)
20
10
20
(10 − 2.7)
10
10
10
(10 − 2.7)
=
7.3 20 10
20 7.3 10
10 10 7.3