In matrix notation
σ
n
ð Þ
n o
¼ λ n
f g
Since
σ
n
ð Þ
n o
¼ n
f g σ
½
Therefore, we can write
n
f g Á σ
½ ¼ λ n
f g
This can be written as
n
f g σ
½ À λ I
½
½
¼0
ð3:3Þ
where [I] is an identity matrix
I ¼
1 0 0
0 1 0
0 0 1
2
6
4
3
7
5
n
f g is a row matrix {n 1 , n 2 , n 3 } where n i defines the direction cosine. λ is the
principal stress values sought. Thus,
λ I
½ ¼
λ 0 0
0 λ 0
0 0 λ
2
6
4
3
7
5
Equation (3.3) has a solution only if the determinant of the matrix is equal to zero
since direction cosines’ vector cannot be zero and because n
2
1 þ n
2
2 þ n
2
3 ¼ 1 must be
satisfied:
Hence,
σ
½ À λ I
½
½
¼0
σ XX À λ
σ XY
σ XZ
σ YX
σ YY À λ
σ YZ
σ ZX
σ ZY
σ ZZ À λ
¼ 0
2.5 Principal Stresses and Principal Axes
23
σ
n
ð Þ
n o
¼ λ n
f g
Since
σ
n
ð Þ
n o
¼ n
f g σ
½
Therefore, we can write
n
f g Á σ
½ ¼ λ n
f g
This can be written as
n
f g σ
½ À λ I
½
½
¼0
ð3:3Þ
where [I] is an identity matrix
I ¼
1 0 0
0 1 0
0 0 1
2
6
4
3
7
5
n
f g is a row matrix {n 1 , n 2 , n 3 } where n i defines the direction cosine. λ is the
principal stress values sought. Thus,
λ I
½ ¼
λ 0 0
0 λ 0
0 0 λ
2
6
4
3
7
5
Equation (3.3) has a solution only if the determinant of the matrix is equal to zero
since direction cosines’ vector cannot be zero and because n
2
1 þ n
2
2 þ n
2
3 ¼ 1 must be
satisfied:
Hence,
σ
½ À λ I
½
½
¼0
σ XX À λ
σ XY
σ XZ
σ YX
σ YY À λ
σ YZ
σ ZX
σ ZY
σ ZZ À λ
¼ 0
2.5 Principal Stresses and Principal Axes
23