34
2 Analysis of Stress
Solution The stress invariants are
I 1 = 4 + 6 + 8 = 18
I 2 = 4 × 6 + 6 × 8 + 4 × 8 − 1 × 1 − 2 × 2 − 0 = 99
I 3 = 4 × 48 − 1 × 8 + 2 × (−12) = 160
The direction cosines for the transformation are given by
x
y
z
x
1
√
2
1
√
2
0
y
−
1
√
2
1
√
2
0
z
0
0
1
Using Eqs. (2.27), (2.27a), (2.27b), (2.27c), (2.27d) and (2.27e), we get
σ x = 4 ×
1
2
+ 6 ×
1
2
+ 0 + 2 × 1 ×
1
2
+ 0 + 0
= 6 MPa
σ y = 4 ×
1
2
+ 6 ×
1
2
+ 0 − 2 × 1 ×
1
2
+ 0 + 0
= 4 MPa
σ z = 0 + 0 + 8 × 1 + 0 + 0 + 0
= 8 MPa