Mechanics of Aircraft structures
C.T. Sun
2.7
Find the principal stresses and corresponding principal directions for the stresses
given in Problem 2.6. Check the result with other methods such as Mohr’s
circle.
Solution:
(a)
The stress given in problem 2.6 is
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
0
0
0
0
3
2
0
2
4
]
[
ij
σ
,
To find the principal stresses, we require that
[
]
[ ]
0
=
−
I
ij
σ
σ
, or
0
0
0
0
3
2
0
2
4
=
−
−
−
σ
σ
σ
Expanding the determinant yields
(
)
0
8
7
2
=
+
−
−
σ
σ
σ
, the solutions of
σ
are
0
=
σ
, or
2
17
7
±
=
σ
, (which are 1.43845 and 5.56155)
--- ANS
(i)
When
0
1
=
σ
We have the equations
⎪
⎩
⎪
⎨
⎧
=
⋅
=
+
=
+
0
0
0
3
2
0
2
4
z
y
x
y
x
n
n
n
n
n
, and also we have
(
)
(
)
(
)
1
2
2
2
=
+
+
z
y
x
n
n
n
So the solutions can be obtained uniquely as
⎪
⎩
⎪
⎨
⎧
=
=
=
1
0
0
z
y
x
n
n
n
, and
is the corresponding principal direction
⎪
⎭
⎪
⎬
⎫
⎪
⎩
⎪
⎨
⎧
=
⎪
⎭
⎪
⎬
⎫
⎪
⎩
⎪
⎨
⎧
1
0
0
)
1
(
z
y
x
n
n
n
--- ANS
(ii)
When
43845
.
1
2
=
σ
We have the equations
⎪
⎩
⎪
⎨
⎧
=
−
=
+
=
+
0
43845
.
1
0
56155
.
1
2
0
2
56155
.
2
z
y
x
y
x
n
n
n
n
n
, and also we have
(
)
(
)
(
)
1
2
2
2
=
+
+
z
y
x
n
n
n
2.7.1

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