86
3 Analysis of Strain
∴ A 1 = 0.046984
B 1 = −
0.15 0.05
−0.04 −0.178
= −[0.15 × (−0.178) + (0.05) (0.04)]
∴ B 1 = 0.0247
C 1 =
0.15 −0.278
−0.04 0.05
= 0.15 × 0.05 − 0.278 × 0.04
∴ C 1 = −0.00362
A
2
1 + B
2
1 + C
2
1 =
(0.046984)
2
+ (0.0247)
2
+ (−0.00362)
2
= 0.0532
∴ l 1 =
A 1
A
2
1 + B
2
1 + C
2
1
=
0.046984
0.0532
= 0.883
m 1 =
B 1
A
2
1 + B
2
1 + C
2
1
=
0.0247
0.0532
= 0.464
n 1 =
C 1
A
2
1 + B
2
1 + C
2
1
=
−0.00362
0.0532
= −0.068
Similarly, the principal directions for ε 2 can be determined as follows:
⎡
⎣
(0.1 + 0.0031)
0.15
−0.04
0.15
(−0.05 + 0.0031)
0.05
−0.04
0.05
(0.05 + 0.0031)
⎤
⎦
=
⎡
⎣
0.1031 0.15 −0.04
0.15 −0.0469 0.05
−0.04 0.05 0.0531
⎤
⎦
A 2 =
−0.0469 0.05
0.05 0.0531
= −0.00249 − 0.0025 = −0.00499
B 2 =
0.15 0.05
−0.04 0.0531
= −(0.007965 + 0.002) = −0.009965
C 2 =
0.15 −0.0469
−0.04 0.05
= 0.0075 − 0.00188 = 0.00562