Page 227 - Analysis and Design of Energy Geostructures

200 Analysis and Design of Energy Geostructures

The characteristic polynomial is defined as follows:
3 2
λ 2 I 1 λ 1 I 2 λ 2 I 3 5 0
The solution of the above equation is found for three real eigenvalues
λ 1 5σ 1 , λ 2 5σ 2 and λ 3 5σ 3 .
That is:
2
3
3
2
λ 1 2 I 1 λ 1 1 I 2 λ 1 2 I 3 5 10 2 17 10 1 80 10 2 100 5 0
3
2
2
3
λ 2 2 I 1 λ 2 1 I 2 λ 2 2 I 3 5 5 2 17 5 1 80 5 2 100 5 0
3 2 3 2
λ 3 2 I 1 λ 3 1 I 2 λ 3 2 I 3 5 2 2 17 2 1 80 2 2 100 5 0
t. i. According to Cauchy’s stress theorem, the component of the stress
vector that acts on this surface are:

2 3
1 1 0
n

t 5 1 2 3 1 2 1 0 5 3 2 1 3
4
5
i
0 0 1
ii. The magnitudes of the normal stresses are the diagonal components
σ kk of the stress tensor that is
σ 11 5 1 kPa
σ 22 52 2 kPa

σ 33 5 1 kPa

The magnitudes of the shear stresses are the off-diagonal compo-
nents σ kl of the stress tensor that is

σ 12 5 1 kPa

σ 21 5 1 kPa

σ 13 5 σ 23 5σ 31 5 σ 32 5 0 kPa